Mathematical Methods
Mathematical Methods
Mathematical Methods
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Hypergeometric function Generalized hypergeometric function SturmLiouville theory Hermite polynomials Jacobi polynomials Legendre polynomials Chebyshev polynomials Gegenbauer polynomials Laguerre polynomials Eigenfunction 1 11 21 27 40 43 49 62 64 73
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Hypergeometric function
Hypergeometric function
In mathematics, the Gaussian or ordinary hypergeometric function 2F1(a,b;c;z) is a special function represented by the hypergeometric series, that includes many other special functions as specific or limiting cases. [It] is a solution of a second-order linear ordinary differential equation (ODE). Every second-order linear ODE with three regular singular points can be transformed into this equation. For systematic lists of some of the many thousands of published identities involving the hypergeometric function, see the reference works by Arthur Erdlyi, Wilhelm Magnus, and Fritz Oberhettinger et al.(1953), Abramowitz & Stegun (1965), and Daalhuis(2010).
History
The term "hypergeometric series" was first used by John Wallis in his 1655 book Arithmetica Infinitorum. Hypergeometric series were studied by Leonhard Euler, but the first full systematic treatment was given by Carl Friedrich Gauss(1813). Studies in the nineteenth century included those of Ernst Kummer(1836), and the fundamental characterisation by Bernhard Riemann(1857) of the hypergeometric function by means of the differential equation it satisfies. Riemann showed that the second-order differential equation for 2F1(z), examined in the complex plane, could be characterised (on the Riemann sphere) by its three regular singularities. The cases where the solutions are algebraic functions were found by Hermann Schwarz (Schwarz's list).
It is undefined (or infinite) if c equals a non-positive integer. Here (q)n is the (rising) Pochhammer symbol, which is defined by:
The series terminates if either a or b is a nonpositive integer. For complex arguments z with |z|1 it can be analytically continued along any path in the complex plane that avoids the branch points 0 and 1. As c goes to a non-positive integer m, 2F1(z) goes to infinity, but if we divide by the gamma function (c), we have a limit:
2 1
F (z) is the most usual type of generalized hypergeometric series pFq, and is often designated simply F(z).
Hypergeometric function
Special cases
Many of the common mathematical functions can be expressed in terms of the hypergeometric function, or as limiting cases of it. Some typical examples are
The confluent hypergeometric function (or Kummer's function) can be given as a limit of the hypergeometric function
so all functions that are essentially special cases of it, such as Bessel functions, can be expressed as limits of hypergeometric functions. These include most of the commonly used functions of mathematical physics. Legendre functions are solutions of a second order differential equation with 3 regular singular points so can be expressed in terms of the hypergeometric function in many ways, for example
Several orthogonal polynomials, including Jacobi polynomials P(,) n and their special cases Legendre polynomials, Chebyshev polynomials, Gegenbauer polynomials can be written in terms of hypergeometric functions using
cases
include
Krawtchouk
polynomials,
Meixner
polynomials,
Elliptic modular functions can sometimes be expressed as the inverse functions of ratios of hypergeometric functions whose arguments a, b, c are 1, 1/2, 1/3, ... or 0. For example, if
then
is an elliptic modular function of . Incomplete beta functions Bx(p,q) are related by The complete elliptic integrals K and E are given by
Hypergeometric function
which has three regular singular points: 0,1 and . The generalization of this equation to three arbitrary regular singular points is given by Riemann's differential equation. Any second order differential equation with three regular singular points can be converted to the hypergeometric differential equation by a change of variables.
If c is a non-positive integer 1m, then the first of these solutions doesn't exist and must be replaced by The second solution doesn't exist when c is an integer greater than 1, and is equal to the first solution, or its replacement, when c is any other integer. So when c is an integer, a more complicated expression must be used for a second solution, equal to the first solution multiplied by ln(z), plus another series in powers of z, involving the digamma function. See Abramowitz & Stegun (1965) for details. Around z=1, if cab is not an integer, one has two independent solutions
and
and
Again, when the conditions of non-integrality are not met, there exist other solutions that are more complicated. Any 3 of the above 6 solutions satisfy a linear relation as the space of solutions is 2-dimensional, giving (6 3) =20 linear relations between them called connection formulas.
Kummer's 24 solutions
A second order Fuchsian equation with n singular points has a group of symmetries acting (projectively) on its solutions, isomorphic to the Coxeter group Dn of order n!2n1. For the hypergeometric equation n=3, so the group is of order 24 and is isomorphic to the symmetric group on 4 points, and was first described by Kummer. The isomorphism with the symmetric group is accidental and has no analogue for more than 3 singular points, and it is sometimes better to think of the group as an extension of the symmetric group on 3 points (acting as permutations of the 3 singular points) by a Klein 4-group (whose elements change the signs of the differences of the exponents at an
Hypergeometric function even number of singular points). Kummer's group of 24 transformations is generated by the three transformations taking a solution F(a,b;c;z) to one of
which correspond to the transpositions (12), (23), and (34) under an isomorphism with the symmetric group on 4 points 1, 2, 3, 4. (The first and third of these are actually equal to F(a,b;c;z) whereas the second is an independent solution to the differential equation.) Applying Kummer's 24=64 transformations to the hypergeometric function gives the 6 = 23 solutions above corresponding to each of the 2 possible exponents at each of the 3 singular points, each of which appears 4 times because of the identities
Q-form
The hypergeometric differential equation may be brought into the Q-form
by making the substitution w = uv and eliminating the first-derivative term. One finds that
is also sometimes used. Note that the connection coefficients become Mbius transformations on the triangle maps. Note that each triangle map is regular at z {0, 1, } respectively, with
and
Hypergeometric function In the special case of , and real, with 0,,<1 then the s-maps are conformal maps of the upper half-plane H to triangles on the Riemann sphere, bounded by circular arcs. This mapping is a special case of a SchwarzChristoffel mapping. The singular points 0,1 and are sent to the triangle vertices. The angles of the triangle are , and respectively. Furthermore, in the case of =1/p, =1/q and =1/r for integers p, q, r, then the triangle tiles the sphere, and the s-maps are inverse functions of automorphic functions for the triangle group p,q,r=(p,q,r).
Monodromy group
The monodromy of a hypergeometric equation describes how fundamental solutions change when analytically continued around paths in the z plane that return to the same point. That is, when the path winds around a singularity of 2F1, the value of the solutions at the endpoint will differ from the starting point. Two fundamental solutions of the hypergeometric equation are related to each other by a linear transformation; thus the monodromy is a mapping (group homomorphism):
where 1 is the fundamental group. In other words the monodromy is a two dimensional linear representation of the fundamental group. The monodromy group of the equation is the image of this map, i.e. the group generated by the monodromy matrices.
Integral formulas
Euler type
If B is the beta function then
provided |z|<1 or |z|=1 and both sides converge, and can be proved by expanding (1zx)a using the binomial theorem and then integrating term by term. This was given by Euler in 1748 and implies Euler's and Pfaff's hypergeometric transformations. Other representations, corresponding to other branches, are given by taking the same integrand, but taking the path of integration to be a closed Pochhammer cycle enclosing the singularities in various orders. Such paths correspond to the monodromy action.
Barnes integral
Barnes used the theory of residues to evaluate the Barnes integral
as
where the contour is drawn to separate the poles 0, 1, 2... from the poles a, a1,..., b, b1,....
Hypergeometric function
John transform
The Gauss hypergeometric function can be written as a John transform (Gelfand, Gindikin & Graev 2003, 2.1.2).
are called contiguous to 2F1(a,b;c;z). Gauss showed that 2F1(a,b;c;z) can be written as a linear combination of any two of its contiguous functions, with rational coefficients in terms of a,b,c, and z. This gives (6 2)=15 relations, given by identifying any two lines on the right hand side of
and so on.
Repeatedly applying these relations gives a linear relation over C(z) between any three functions of the form
Hypergeometric function
Transformation formulas
Transformation formulas relate two hypergeometric functions at different values of the argument z.
which in turn follow from Euler's integral representation. For extension of Euler's first and second transformations, see papers by Rathie & Paris and Rakha & Rathie.
Quadratic transformations
If two of the numbers 1c, c1, ab, ba, a+bc, cab are equal or one of them is 1/2 then there is a quadratic transformation of the hypergeometric function, connecting it to a different value of z related by a quadratic equation. The first examples were given by Kummer (1836), and a complete list was given by Goursat (1881). A typical example is
There are also some transformations of degree 4 and 6. Transformations of other degrees only exist if a, b, and c are certain rational numbers.
which follows from Euler's integral formula by putting z=1. It includes the Vandermonde identity, first found by Zhu Shijie (=Chu Shi-Chieh), as a special case. Dougall's formula generalizes this to the bilateral hypergeometric series at z=1.
Hypergeometric function
and Gauss's theorem by putting z=1 in the first identity. For generalization of Kummer's summation, see a paper by Lavoie, et al.
Values at z=1/2
Gauss's second summation theorem is
Bailey's theorem is
For generalizations of Gauss's second summation theorem and Bailey's summation theorem, see a paper by Lavoie, et al.
Other points
There are many other formulas giving the hypergeometric function as an algebraic number at special rational values of the parameters, some of which are listed in (Gessel & Stanton 1982) and (Koepf 1995). Some typical examples are given by
References
[1] Hille, Einar (1976), Ordinary differential equations in the complex domain, Dover, pp. 374401, ISBN 0-486-69620-0, Chapter 10, "The Schwarzian".
Abramowitz, Milton; Stegun, Irene A., eds. (1965), "Chapter 15" (http://www.math.sfu.ca/~cbm/aands/ page_555.htm), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover, p.555, ISBN978-0486612720, MR 0167642 (http://www.ams.org/ mathscinet-getitem?mr=0167642). Andrews, George E.; Askey, Richard & Roy, Ranjan (1999). Special functions. Encyclopedia of Mathematics and its Applications 71. Cambridge University Press. ISBN978-0-521-62321-6. MR 1688958 (http://www.ams.
Hypergeometric function org/mathscinet-getitem?mr=1688958). Bailey, W.N. (1935). Generalized Hypergeometric Series. Cambridge. Beukers, Frits (2002), Gauss' hypergeometric function (http://www.math.uu.nl/people/beukers/ MRIcourse93.ps). (lecture notes reviewing basics, as well as triangle maps and monodromy) Daalhuis, Adri B. Olde (2010), "Hypergeometric function" (http://dlmf.nist.gov/15), in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN978-0521192255, MR 2723248 (http://www.ams.org/ mathscinet-getitem?mr=2723248) Erdlyi, Arthur; Magnus, Wilhelm; Oberhettinger, Fritz & Tricomi, Francesco G. (1953). Higher transcendental functions (http://apps.nrbook.com/bateman/Vol1.pdf). Vol. I. New YorkTorontoLondon: McGrawHill Book Company, Inc. ISBN978-0-89874-206-0. MR 0058756 (http://www.ams.org/ mathscinet-getitem?mr=0058756). Gasper, George & Rahman, Mizan (2004). Basic Hypergeometric Series, 2nd Edition, Encyclopedia of Mathematics and Its Applications, 96, Cambridge University Press, Cambridge. ISBN 0-521-83357-4.
Gauss, Carl Friedrich (1813). "Disquisitiones generales circa seriem infinitam " (http://books.google.com/books?id=uDMAAAAAQAAJ). Commentationes societatis regiae scientarum Gottingensis recentiores (in Latin) (Gttingen) 2. (a reprint of this paper can be found in Carl Friedrich Gauss, Werke (http://books.google.com/books?id=uDMAAAAAQAAJ), p.125) Gelfand, I. M.; Gindikin, S.G. & Graev, M.I. (2003) [2000]. Selected topics in integral geometry (http://books. google.com/books?isbn=0821829327). Translations of Mathematical Monographs 220. Providence, R.I.: American Mathematical Society. ISBN978-0-8218-2932-5. MR 2000133 (http://www.ams.org/ mathscinet-getitem?mr=2000133). Gessel, Ira & Stanton, Dennis (1982). "Strange evaluations of hypergeometric series". SIAM Journal on Mathematical Analysis 13 (2): 295308. doi: 10.1137/0513021 (http://dx.doi.org/10.1137/0513021). ISSN 0036-1410 (http://www.worldcat.org/issn/0036-1410). MR 647127 (http://www.ams.org/ mathscinet-getitem?mr=647127). Goursat, douard (1881). "Sur l'quation diffrentielle linaire, qui admet pour intgrale la srie hypergomtrique" (http://www.numdam.org/item?id=ASENS_1881_2_10__S3_0). Annales Scientifiques de l'cole Normale Suprieure (in French) 10: 3142. Retrieved 2008-10-16. Heckman, Gerrit & Schlichtkrull, Henrik (1994). Harmonic Analysis and Special Functions on Symmetric Spaces. San Diego: Academic Press. ISBN0-12-336170-2. (part 1 treats hypergeometric functions on Lie groups) Klein, Felix (1981). Vorlesungen ber die hypergeometrische Funktion (http://resolver.sub.uni-goettingen.de/ purl?PPN375394591). Grundlehren der Mathematischen Wissenschaften (in German) 39. Berlin, New York: Springer-Verlag. ISBN978-3-540-10455-1. MR 668700 (http://www.ams.org/ mathscinet-getitem?mr=668700). Koepf, Wolfram (1995). "Algorithms for m-fold hypergeometric summation". Journal of Symbolic Computation 20 (4): 399417. doi: 10.1006/jsco.1995.1056 (http://dx.doi.org/10.1006/jsco.1995.1056). ISSN 0747-7171 (http://www.worldcat.org/issn/0747-7171). MR 1384455 (http://www.ams.org/ mathscinet-getitem?mr=1384455). Kummer, Ernst Eduard (1836). "ber die hypergeometrische Reihe " (http://resolver.sub.uni-goettingen.
de/purl?GDZPPN00214056X). Journal fr die reine und angewandte Mathematik (in German) 15: 3983, 127172. ISSN 0075-4102 (http://www.worldcat.org/issn/0075-4102). Lavoie,J.L., Grondin, F. and Rathie, A.K., Generalizations of Whipple's theorem on the sum of a 3F2, J. Comput. Appl. Math., 72, 293-300,(1996).
Hypergeometric function Press, W.H.; Teukolsky, S.A.; Vetterling, W.T. & Flannery, B.P. (2007). "Section 6.13. Hypergeometric Functions" (http://apps.nrbook.com/empanel/index.html#pg=318). Numerical Recipes: The Art of Scientific Computing (3rd ed.). New York: Cambridge University Press. ISBN978-0-521-88068-8. Riemann, Bernhard (1857). "Beitrge zur Theorie der durch die Gauss'sche Reihe F(, , , x) darstellbaren Functionen" (http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=GDZPPN002018691). Abhandlungen der Mathematicshen Classe der K oniglichen Gesellschaft der Wissenschaften zu G ottingen (Gttingen: Verlag der Dieterichschen Buchhandlung) 7: 322. (a reprint of this paper can be found in All publications of Riemann (http://www.emis.de/classics/Riemann/PFunct.pdf)PDF) Slater, Lucy Joan (1960). Confluent hypergeometric functions. Cambridge, UK: Cambridge University Press. MR 0107026 (http://www.ams.org/mathscinet-getitem?mr=0107026). Slater, Lucy Joan (1966). Generalized hypergeometric functions. Cambridge, UK: Cambridge University Press. ISBN0-521-06483-X. MR 0201688 (http://www.ams.org/mathscinet-getitem?mr=0201688). (there is a 2008 paperback with ISBN 978-0-521-09061-2) Wall, H.S. (1948). Analytic Theory of Continued Fractions. D. Van Nostrand Company, Inc. Whittaker, E.T. & Watson, G.N. (1927). A Course of Modern Analysis. Cambridge, UK: Cambridge University Press. Yoshida, Masaaki (1997). Hypergeometric Functions, My Love: Modular Interpretations of Configuration Spaces. Braunschweig/Wiesbaden: Friedr. Vieweg & Sohn. ISBN3-528-06925-2. MR 1453580 (http://www. ams.org/mathscinet-getitem?mr=1453580). Rathie, Arjun K. & Paris, R.B.: An extension of the Euler's-type transformation for the 3F2 series: Far East J.Math.Sci., 27(1), 43-48 (2007). Rakha, M.A. & Rathie, Arjun K. : Extensions of Euler's type- II transformation and Saalschutz's theorem: Bull. Korean Math. Soc.,48(1), 151-156 (2011).
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External links
Hazewinkel, Michiel, ed. (2001), "Hypergeometric function" (http://www.encyclopediaofmath.org/index. php?title=p/h048450), Encyclopedia of Mathematics, Springer, ISBN978-1-55608-010-4 John Pearson, Computation of Hypergeometric Functions (http://people.maths.ox.ac.uk/porterm/research/ pearson_final.pdf) (University of Oxford, MSc Thesis) Marko Petkovsek, Herbert Wilf and Doron Zeilberger, The book "A = B" (http://www.cis.upenn.edu/~wilf/ AeqB.html) (freely downloadable) Weisstein, Eric W., " Hypergeometric Function (http://mathworld.wolfram.com/HypergeometricFunction. html)", MathWorld.
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Notation
A hypergeometric series is formally defined as a power series
where A(n) and B(n) are polynomials in n. For example, in the case of the series for the exponential function, , n=n!1 and n+1/n=1/(n+1). So this satisfies the definition with A(n)=1 and B(n)=n+1. It is customary to factor out the leading term, so 0 is assumed to be 1. The polynomials can be factored into linear factors of the form (aj+n) and (bk+n) respectively, where the aj and bk are complex numbers. For historical reasons, it is assumed that (1+n) is a factor of B. If this is not already the case then both A and B can be multiplied by this factor; the factor cancels so the terms are unchanged and there is no loss of generality. The ratio between consecutive coefficients now has the form , where c and d are the leading coefficients of A and B. The series then has the form , or, by scaling z by the appropriate factor and rearranging, . This has the form of an exponential generating function. The standard notation for this series is or Using the rising factorial or Pochhammer symbol:
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(Note that this use of the Pochhammer symbol is not standard, however it is the standard usage in this context.)
Special cases
Many of the special functions in mathematics are special cases of the confluent hypergeometric function or the hypergeometric function; see the corresponding articles for examples. Some of the functions related to more complicated hypergeometric functions include: Dilogarithm:
Hahn polynomials:
Wilson polynomials:
Terminology
When all the terms of the series are defined and it has a non-zero radius of convergence, then the series defines an analytic function. Such a function, and its analytic continuations, is called the hypergeometric function. The case when the radius of convergence is 0 yields many interesting series in mathematics, for example the incomplete gamma function has the asymptotic expansion
which could be written za1ez2F0(1a,1;;z1). However, the use of the term hypergeometric series is usually restricted to the case where the series defines an actual analytic function. The ordinary hypergeometric series should not be confused with the basic hypergeometric series, which, despite its name, is a rather more complicated and recondite series. The "basic" series is the q-analog of the ordinary hypergeometric series. There are several such generalizations of the ordinary hypergeometric series, including the ones coming from zonal spherical functions on Riemannian symmetric spaces. The series without the factor of n! in the denominator (summed over all integers n, including negative) is called the bilateral hypergeometric series.
Convergence conditions
There are certain values of the aj and bk for which the numerator or the denominator of the coefficients is 0. If any aj is a non-positive integer (0, 1, 2, etc.) then the series only has a finite number of terms and is, in fact, a polynomial of degree aj. If any bk is a non-positive integer (excepting the previous case with bk < aj) then the denominators become 0 and the series is undefined. Excluding these cases, the ratio test can be applied to determine the radius of convergence. If p < q + 1 then the ratio of coefficients tends to zero. This implies that the series converges for any finite value of z. An example is the power series for the exponential function.
Generalized hypergeometric function If p = q + 1 then the ratio of coefficients tends to one. This implies that the series converges for |z|<1 and diverges for |z|>1. Whether it converges for |z|=1 is more difficult to determine. Analytic continuation can be employed for larger values of z. If p > q + 1 then the ratio of coefficients grows without bound. This implies that, besides z=0, the series diverges. This is then a divergent or asymptotic series, or it can be interpreted as a symbolic shorthand for a differential equation that the sum satisfies. The question of convergence for p=q+1 when z is on the unit circle is more difficult. It can be shown that the series converges absolutely at z = 1 if . Further, if p=q+1, and z is real, then the following convergence result holds (Quigley et al 2013):
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Basic properties
It is immediate from the definition that the order of the parameters aj, or the order of the parameters bk can be changed without changing the value of the function. Also, if any of the parameters aj is equal to any of the parameters bk, then the matching parameters can be "cancelled out", with certain exceptions when the parameters are non-positive integers. For example, .
Differentiation
The generalized hypergeometric function satisfies
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From the differentiation formulas given above, the linear space spanned by
contains each of
Since the space has dimension 2, any three of these p+q+2 functions are linearly dependent. These dependencies can be written out to generate a large number of identities involving . For example, in the simplest non-trivial case, , , , So . This, and other important examples, , ,
, , , can be used to generate continued fraction expressions known as Gauss's continued fraction. Similarly, by applying the differentiation formulas twice, there are such functions contained in
which has dimension three so any four are linearly dependent. This generates more identities and the process can be continued. The identities thus generated can be combined with each other to produce new ones in a different way. A function obtained by adding 1 to exactly one of the parameters aj, bk in is called contiguous to
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Using the technique outlined above, an identity relating six identities relating
and any two of its four contiguous functions, and fifteen identities relating
and any two of its six contiguous functions have been found. (The first one was derived in the previous paragraph. The last fifteen were given by Gauss in his 1812 paper.)
Identities
A number of other hypergeometric function identities were discovered in the nineteenth and twentieth centuries.
Saalschtz's theorem
Saalschtz's theorem[1] (Saalschtz 1890) is
For extension of this theorem, see a research paper by Rakha & Rathie.
Dixon's identity
Dixon's identity,[2] first proved by Dixon (1902), gives the sum of a well-poised 3F2 at 1:
Dougall's formula
Dougall's formula (Dougall1907) gives the sum of a terminating well-poised [3] series:
provided that m is a non-negative integer (so that the series terminates) and
Many of the other formulas for special values of hypergeometric functions can be derived from this as special or limiting cases.
where ; Identity 2.
which links Bessel functions to 2F2; this reduces to Kummer's second formula for b = 2a:
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Kummer's relation
Kummer's relation is
Clausen's formula
Clausen's formula
Special cases
The series 0F0
As noted earlier, where k is a constant. . The differential equation for this function is , which has solutions
or
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or
where k, l are constants. (If a is a positive integer, the independent solution is given by the appropriate Bessel function of the second kind.)
or
where k, l are constants. When a is a non-positive integer, n, is a polynomial. Up to constant factors, these are the Laguerre
polynomials. This implies Hermite polynomials can be expressed in terms of 1F1 as well.
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or
It is known as the hypergeometric differential equation. When c is not a positive integer, the substitution
where k, l are constants. Different solutions can be derived for other values of z. In fact there are 24 solutions, known as the Kummer solutions, derivable using various identities, valid in different regions of the complex plane. When a is a non-positive integer, n,
is a polynomial. Up to constant factors and scaling, these are the Jacobi polynomials. Several other classes of orthogonal polynomials, up to constant factors, are special cases of Jacobi polynomials, so these can be expressed using 2F1 as well. This includes Legendre polynomials and Chebyshev polynomials. A wide range of integrals of elementary functions can be expressed using the hypergeometric function, e.g.:
Generalizations
The generalized hypergeometric function is linked to the Meijer G-function and the MacRobert E-function. Hypergeometric series were generalised to several variables, for example by Paul Emile Appell; but a comparable general theory took long to emerge. Many identities were found, some quite remarkable. A generalization, the q-series analogues, called the basic hypergeometric series, were given by Eduard Heine in the late nineteenth century. Here, the ratios considered of successive terms, instead of a rational function of n, are a rational function of qn. Another generalization, the elliptic hypergeometric series, are those series where the ratio of terms is an elliptic function (a doubly periodic meromorphic function) of n.
Generalized hypergeometric function During the twentieth century this was a fruitful area of combinatorial mathematics, with numerous connections to other fields. There are a number of new definitions of general hypergeometric functions, by Aomoto, Israel Gelfand and others; and applications for example to the combinatorics of arranging a number of hyperplanes in complex N-space (see arrangement of hyperplanes). Special hypergeometric functions occur as zonal spherical functions on Riemannian symmetric spaces and semi-simple Lie groups. Their importance and role can be understood through the following example: the hypergeometric series 2F1 has the Legendre polynomials as a special case, and when considered in the form of spherical harmonics, these polynomials reflect, in a certain sense, the symmetry properties of the two-sphere or, equivalently, the rotations given by the Lie group SO(3). In tensor product decompositions of concrete representations of this group Clebsch-Gordan coefficients are met, which can be written as 3F2 hypergeometric series. Bilateral hypergeometric series are a generalization of hypergeometric functions where one sums over all integers, not just the positive ones. FoxWright functions are a generalization of generalized hypergeometric functions where the Pochhammer symbols in the series expression are generalised to gamma functions of linear expressions in the index n.
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Citations
[1] See or for a proof. [2] See for a detailed proof. An alternative proof is in [3] http:/ / mathworld. wolfram. com/ Well-Poised. html
References
Askey, R. A.; Daalhuis, Adri B. Olde (2010), "Generalized hypergeometric function" (http://dlmf.nist.gov/16), in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN978-0521192255, MR 2723248 (http://www.ams. org/mathscinet-getitem?mr=2723248) Andrews, George E.; Askey, Richard & Roy, Ranjan (1999). Special functions. Encyclopedia of Mathematics and its Applications 71. Cambridge University Press. ISBN978-0-521-62321-6; 978-0-521-78988-2 Check |isbn= value (help). MR 1688958 (http://www.ams.org/mathscinet-getitem?mr=1688958). Bailey, W.N. (1935). Generalized Hypergeometric Series. Cambridge Tracts in Mathematics and Mathematical Physics 32. London: Cambridge University Press. Zbl 0011.02303 (http://www.zentralblatt-math.org/zmath/ en/search/?format=complete&q=an:0011.02303). Dixon, A.C. (1902). "Summation of a certain series". Proc. London Math. Soc. 35 (1): 284291. doi: 10.1112/plms/s1-35.1.284 (http://dx.doi.org/10.1112/plms/s1-35.1.284). JFM 34.0490.02 (http://www. zentralblatt-math.org/zmath/en/search/?format=complete&q=an:34.0490.02). Dougall, J. (1907). "On Vandermonde's theorem and some more general expansions". Proc. Edinburgh Math. Soc. 25: 114132. doi: 10.1017/S0013091500033642 (http://dx.doi.org/10.1017/S0013091500033642). Gasper, George; Rahman, Mizan (2004). Basic Hypergeometric Series. Encyclopedia of Mathematics and Its Applications 96 (2nd ed.). Cambridge, UK: Cambridge University Press. ISBN0-521-83357-4. MR 2128719 (http://www.ams.org/mathscinet-getitem?mr=2128719). Zbl 1129.33005 (http://www.zentralblatt-math.org/ zmath/en/search/?format=complete&q=an:1129.33005). (the first edition has ISBN 0-521-35049-2) Gauss, Carl Friedrich (1813). "Disquisitiones generales circa seriam infinitam " (http://books.google.com/books?id=uDMAAAAAQAAJ). Commentationes societatis regiae scientarum Gottingensis recentiores (in Latin) (Gttingen) 2. (a reprint of this paper can be found in Carl Friedrich Gauss, Werke (http://books.google.com/books?id=uDMAAAAAQAAJ), p.125)
Generalized hypergeometric function Heckman, Gerrit & Schlichtkrull, Henrik (1994). Harmonic Analysis and Special Functions on Symmetric Spaces. San Diego: Academic Press. ISBN0-12-336170-2. (part 1 treats hypergeometric functions on Lie groups) Lavoie, J.L.; Grondin, F.; Rathie, A.K.; Arora, K. (1994). "Generalizations of Dixon's theorem on the sum of a 3F2". Math. Comp. 62: 267276. Miller, A. R.; Paris, R. B. (2011). "Euler-type transformations for the generalized hypergeometric function F ". Zeit. Angew. Math. Physik: 3145. doi: 10.1007/s00033-010-0085-0 (http://dx.doi.org/10.1007/ r+2 r+1 s00033-010-0085-0). Quigley, J.; Wilson, K.J.; Walls, L.; Bedford, T. (2013). "A Bayes linear Bayes Method for Estimation of Correlated Event Rates". Risk Analysis. doi: 10.1111/risa.12035 (http://dx.doi.org/10.1111/risa.12035). Rathie, Arjun K.; Pogny, Tibor K. (2008). "New summation formula for 3F2(1/2) and a Kummer-type II transformation of 2F2(x)" (http://hrcak.srce.hr/file/37118). Mathematical Communications 13: 6366. MR 2422088 (http://www.ams.org/mathscinet-getitem?mr=2422088). Zbl 1146.33002 (http://www. zentralblatt-math.org/zmath/en/search/?format=complete&q=an:1146.33002). Rakha, M.A.; Rathie, Arjun K. (2011). "Extensions of Euler's type- II transformation and Saalschutz's theorem". Bull. Korean Math. Soc. 48 (1): 151156. Saalschtz, L. (1890). "Eine Summationsformel". Zeitschrift fr Mathematik und Physik (in German) 35: 186188. JFM 22.0262.03 (http://www.zentralblatt-math.org/zmath/en/search/?format=complete&q=an:22. 0262.03). Slater, Lucy Joan (1966). Generalized Hypergeometric Functions. Cambridge, UK: Cambridge University Press. ISBN0-521-06483-X. MR 0201688 (http://www.ams.org/mathscinet-getitem?mr=0201688). Zbl 0135.28101 (http://www.zentralblatt-math.org/zmath/en/search/?format=complete&q=an:0135.28101). (there is a 2008 paperback with ISBN 978-0-521-09061-2) Yoshida, Masaaki (1997). Hypergeometric Functions, My Love: Modular Interpretations of Configuration Spaces. Braunschweig/Wiesbaden: Friedr. Vieweg & Sohn. ISBN3-528-06925-2. MR 1453580 (http://www. ams.org/mathscinet-getitem?mr=1453580).
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External links
The book "A = B" (http://www.cis.upenn.edu/~wilf/AeqB.html), this book is freely downloadable from the internet. MathWorld Weisstein, Eric W., " Generalized Hypergeometric Function (http://mathworld.wolfram.com/ GeneralizedHypergeometricFunction.html)", MathWorld. Weisstein, Eric W., " Hypergeometric Function (http://mathworld.wolfram.com/HypergeometricFunction. html)", MathWorld. Weisstein, Eric W., " Confluent Hypergeometric Function of the First Kind (http://mathworld.wolfram.com/ ConfluentHypergeometricFunctionoftheFirstKind.html)", MathWorld. Weisstein, Eric W., " Confluent Hypergeometric Limit Function (http://mathworld.wolfram.com/ ConfluentHypergeometricLimitFunction.html)", MathWorld.
SturmLiouville theory
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SturmLiouville theory
In mathematics and its applications, a classical SturmLiouville equation, named after Jacques Charles Franois Sturm (18031855) and Joseph Liouville (18091882), is a real second-order linear differential equation of the form
(1)
where y is a function of the free variable x. Here the functions p(x)>0, q(x), and w(x)>0 are specified at the outset. In the simplest of cases all coefficients are continuous on the finite closed interval [a,b], and p has continuous derivative. In this simplest of all cases, this function "y" is called a solution if it is continuously differentiable on (a,b) and satisfies the equation (1) at every point in (a,b). In addition, the unknown function y is typically required to satisfy some boundary conditions at a and b. The function w(x), which is sometimes called r(x), is called the "weight" or "density" function. The value of is not specified in the equation; finding the values of for which there exists a non-trivial solution of (1) satisfying the boundary conditions is part of the problem called the SturmLiouville (SL) problem. Such values of when they exist are called the eigenvalues of the boundary value problem defined by (1) and the prescribed set of boundary conditions. The corresponding solutions (for such a ) are the eigenfunctions of this problem. Under normal assumptions on the coefficient functions p(x), q(x), and w(x) above, they induce a Hermitian differential operator in some function space defined by boundary conditions. The resulting theory of the existence and asymptotic behavior of the eigenvalues, the corresponding qualitative theory of the eigenfunctions and their completeness in a suitable function space became known as SturmLiouville theory. This theory is important in applied mathematics, where SL problems occur very commonly, particularly when dealing with linear partial differential equations that are separable. A SturmLiouville (SL) problem is said to be regular if p(x), w(x)>0, and p(x), p'(x), q(x), and w(x) are continuous functions over the finite interval [a,b], and have separated boundary conditions of the form
(2)
(3)
Under the assumption that the SL problem is regular, the main tenet of SturmLiouville theory states that: The eigenvalues 1, 2, 3, ... of the regular SturmLiouville problem (1)-(2)-(3) are real and can be ordered such that
Corresponding to each eigenvalue n is a unique (up to a normalization constant) eigenfunction yn(x) which has exactly n1 zeros in (a,b). The eigenfunction yn(x) is called the n-th fundamental solution satisfying the regular SturmLiouville problem (1)-(2)-(3). The normalized eigenfunctions form an orthonormal basis
in the Hilbert space L2([a,b], w(x)dx). Here mn is a Kronecker delta. Note that, unless p(x) is continuously differentiable and q(x), w(x) are continuous, the equation has to be understood in a weak sense.
SturmLiouville theory
22
SturmLiouville form
The differential equation (1) is said to be in SturmLiouville form or self-adjoint form. All second-order linear ordinary differential equations can be recast in the form on the left-hand side of (1) by multiplying both sides of the equation by an appropriate integrating factor (although the same is not true of second-order partial differential equations, or if y is a vector.)
Examples
The Bessel equation:
The Legendre equation: can easily be put into SturmLiouville form, since D(1x2) = 2x, so, the Legendre equation is equivalent to
gives
or, explicitly,
SturmLiouville theory
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can be viewed as a linear operator mapping a function u to another function Lu. We may study this linear operator in the context of functional analysis. In fact, equation (1) can be written as
This is precisely the eigenvalue problem; that is, we are trying to find the eigenvalues 1, 2, 3, ... and the corresponding eigenvectors u1, u2, u3, ... of the L operator. The proper setting for this problem is the Hilbert space L2([a, b], w(x) dx) with scalar product
In this space L is defined on sufficiently smooth functions which satisfy the above boundary conditions. Moreover, L gives rise to a self-adjoint operator. This can be seen formally by using integration by parts twice, where the boundary terms vanish by virtue of the boundary conditions. It then follows that the eigenvalues of a SturmLiouville operator are real and that eigenfunctions of L corresponding to different eigenvalues are orthogonal. However, this operator is unbounded and hence existence of an orthonormal basis of eigenfunctions is not evident. To overcome this problem one looks at the resolvent
where z is chosen to be some real number which is not an eigenvalue. Then, computing the resolvent amounts to solving the inhomogeneous equation, which can be done using the variation of parameters formula. This shows that the resolvent is an integral operator with a continuous symmetric kernel (the Green's function of the problem). As a consequence of the ArzelAscoli theorem this integral operator is compact and existence of a sequence of eigenvalues n which converge to 0 and eigenfunctions which form an orthonormal basis follows from the spectral theorem for compact operators. Finally, note that is equivalent to . If the interval is unbounded, or if the coefficients have singularities at the boundary points, one calls L singular. In this case the spectrum does no longer consist of eigenvalues alone and can contain a continuous component. There is still an associated eigenfunction expansion (similar to Fourier series versus Fourier transform). This is important in quantum mechanics, since the one-dimensional time-independent Schrdinger equation is a special case of a SL equation.
Example
We wish to find a function u(x) which solves the following SturmLiouville problem:
(4)
where the unknowns are and u(x). As above, we must add boundary conditions, we take for example
Observe that if k is any integer, then the function is a solution with eigenvalue = k2. We know that the solutions of a SL problem form an orthogonal basis, and we know from Fourier series that this set of sinusoidal functions is an orthogonal basis. Since orthogonal bases are always maximal (by definition) we conclude that the SL problem in this case has no other eigenvectors.
SturmLiouville theory Given the preceding, let us now solve the inhomogeneous problem
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with the same boundary conditions. In this case, we must write f(x) = x in a Fourier series. The reader may check, either by integrating exp(ikx)xdx or by consulting a table of Fourier transforms, that we thus obtain
This particular Fourier series is troublesome because of its poor convergence properties. It is not clear a priori whether the series converges pointwise. Because of Fourier analysis, since the Fourier coefficients are "square-summable", the Fourier series converges in L2 which is all we need for this particular theory to function. We mention for the interested reader that in this case we may rely on a result which says that Fourier's series converges at every point of differentiability, and at jump points (the function x, considered as a periodic function, has a jump at ) converges to the average of the left and right limits (see convergence of Fourier series). Therefore, by using formula (4), we obtain that the solution is
In this case, we could have found the answer using antidifferentiation. This technique yields u =(x32x)/6, whose Fourier series agrees with the solution we found. The antidifferentiation technique is no longer useful in most cases when the differential equation is in many variables.
The method of separation of variables suggests looking first for solutions of the simple form W = X(x) Y(y) T(t). For such a function W the partial differential equation becomes X"/X + Y"/Y = (1/c2)T"/T. Since the three terms of this equation are functions of x,y,t separately, they must be constants. For example, the first term gives X"=X for a constant . The boundary conditions ("held in a rectangular frame") are W=0 when x=0, L1 or y = 0, L2 and define the simplest possible SL eigenvalue problems as in the example, yielding the "normal mode solutions" for W with harmonic time dependence,
where m and n are non-zero integers, Amn are arbitrary constants, and
The functions Wmn form a basis for the Hilbert space of (generalized) solutions of the wave equation; that is, an arbitrary solution W can be decomposed into a sum of these modes, which vibrate at their individual frequencies . This representation may require a convergent infinite sum.
SturmLiouville theory
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is a solution of the same equation and is linearly independent from y. Further, all solutions are linear combinations of these two solutions. In the SPPS algorithm, one must begin with an arbitrary value 0* (often 0*=0; it does not need to be an eigenvalue) and any solution y0 of (1) with =0* which does not vanish on [a,b]. (Discussion below of ways to find appropriate y0 and 0*.) Two sequences of functions X(n)(t), X~(n)(t) on [a,b], referred to as iterated integrals, are defined recursively as follows. First when n=0, they are taken to be identically equal to 1 on [a,b]. To obtain the next functions they are multiplied alternately by 1/(py02) and wy02 and integrated, specifically
for n odd, for n even, (5)
for n odd,
for n even,
(6)
when n>0. The resulting iterated integrals are now applied as coefficients in the following two power series in : and Then for any (real or complex), u0 and u1 are linearly independent solutions of the corresponding equation (1). (The functions p(x) and q(x) take part in this construction through their influence on the choice of y0.) Next one chooses coefficients c0, c1 so that the combination y = c0u0 + c1u1 satisfies the first boundary condition (2). This is simple to do since X(n)(a)=0 and X~(n)(a)=0, for n>0. The values of X(n)(b) and X~(n)(b) provide the values of u0(b) and u1(b) and the derivatives u0'(b) and u1'(b), so the second boundary condition (3) becomes an equation in a power series in . For numerical work one may truncate this series to a finite number of terms, producing a calculable polynomial in whose roots are approximations of the sought-after eigenvalues. When = 0, this reduces to the original construction described above for a solution linearly independent to a given one. The representations (5),(6) also have theoretical applications in SturmLiouville theory.
SturmLiouville theory value 0=0. In practice if (1) has real coefficients, the solutions based on y0 will have very small imaginary parts which must be discarded.
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Application to PDEs
For a linear second order in one spatial dimension and first order in time of the form:
The first of these equations must be solved as a SturmLiouville problem. Since there is no general analytic (exact) solution to SturmLiouville problems, we can assume we already have the solution to this problem, that is, we have the eigenfunctions and eigenvalues . The second of these equations can be analytically solved once the eigenvalues are known.
Where:
SturmLiouville theory
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References
[1] J. D. Pryce, Numerical Solution of SturmLiouville Problems, Clarendon Press, Oxford, 1993. [2] V. Ledoux, M. Van Daele, G. Vanden Berghe, "Efficient computation of high index SturmLiouville eigenvalues for problems in physics," Comput. Phys. Comm. 180, 2009, 532554. [3] V. V. Kravchenko, R. M. Porter, "Spectral parameter power series for SturmLiouville problems," Mathematical Methods in the Applied Sciences (MMAS) 33, 2010, 459468
Further reading
Hazewinkel, Michiel, ed. (2001), "Sturm-Liouville theory" (http://www.encyclopediaofmath.org/index. php?title=p/s130620), Encyclopedia of Mathematics, Springer, ISBN978-1-55608-010-4 Hartman, Peter (2002). Ordinary Differential Equations (2 ed.). Philadelphia: SIAM. ISBN978-0-89871-510-1. Polyanin, A. D. and Zaitsev, V. F. (2003). Handbook of Exact Solutions for Ordinary Differential Equations (2 ed.). Boca Raton: Chapman & Hall/CRC Press. ISBN1-58488-297-2. Teschl, Gerald (2012). Ordinary Differential Equations and Dynamical Systems (http://www.mat.univie.ac.at/ ~gerald/ftp/book-ode/). Providence: American Mathematical Society. ISBN978-0-8218-8328-0. (Chapter 5) Teschl, Gerald (2009). Mathematical Methods in Quantum Mechanics; With Applications to Schrdinger Operators (http://www.mat.univie.ac.at/~gerald/ftp/book-schroe/). Providence: American Mathematical Society. ISBN978-0-8218-4660-5. (see Chapter 9 for singular SL operators and connections with quantum mechanics) Zettl, Anton (2005). SturmLiouville Theory. Providence: American Mathematical Society. ISBN0-8218-3905-5.
Hermite polynomials
In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence that arise in probability, such as the Edgeworth series; in combinatorics, as an example of an Appell sequence, obeying the umbral calculus; in numerical analysis as Gaussian quadrature; in finite element methods as shape functions for beams; and in physics, where they give rise to the eigenstates of the quantum harmonic oscillator. They are also used in systems theory in connection with nonlinear operations on Gaussian noise. They are named after Charles Hermite (1864)[1] although they were studied earlier by Laplace (1810) and Chebyshev (1859).[2]
Definition
There are two different ways of standardizing the Hermite polynomials:
(the "physicists' Hermite polynomials"). These two definitions are not exactly identical; each one is a rescaling of the other,
These are Hermite polynomial sequences of different variances; see the material on variances below. The notation He and H is that used in the standard references Tom H. Koornwinder, Roderick S. C. Wong, and Roelof Koekoek et al.(2010) and Abramowitz & Stegun. The polynomials Hen are sometimes denoted by Hn,
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is the probability density function for the normal distribution with expected value 0 and standard deviation 1. The first eleven probabilists' Hermite polynomials are:
Hermite polynomials
29
Properties
Hn is a polynomial of degree n. The probabilists' version He has leading coefficient 1, while the physicists' version H has leading coefficient 2n.
Orthogonality
Hn(x) and Hen(x) are nth-degree polynomials for n= 0, 1, 2, 3,.... These polynomials are orthogonal with respect to the weight function (measure) (He) or (H) i.e., we have
when m n. Furthermore, (probabilist) or (physicist). The probabilist polynomials are thus orthogonal with respect to the standard normal probability density function.
Completeness
The Hermite polynomials (probabilist or physicist) form an orthogonal basis of the Hilbert space of functions satisfying
in which the inner product is given by the integral including the Gaussian weight function w(x) defined in the preceding section,
An orthogonal basis for L2(R,w(x)dx) is a complete orthogonal system. For an orthogonal system, completeness is equivalent to the fact that the 0 function is the only function L2(R,w(x)dx) orthogonal to all functions in the system. Since the linear span of Hermite polynomials is the space of all polynomials, one has to show (in physicist case) that if satisfies
for every n0, then =0. One possible way to do it is to see that the entire function
vanishes identically. The fact that F(it)=0 for every t real means that the Fourier transform of (x)exp(x2) is 0, hence is 0 almost everywhere. Variants of the above completeness proof apply to other weights with exponential decay. In the Hermite case, it is also possible to prove an explicit identity that implies completeness (see
Hermite polynomials "Completeness relation" below). An equivalent formulation of the fact that Hermite polynomials are an orthogonal basis for L2(R,w(x)dx) consists in introducing Hermite functions (see below), and in saying that the Hermite functions are an orthonormal basis for L2(R).
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where is a constant, with the boundary conditions that u should be polynomially bounded at infinity. With these boundary conditions, the equation has solutions only if is a non-negative integer, and up to an overall scaling, the solution is uniquely given by u(x)=He(x). Rewriting the differential equation as an eigenvalue problem solutions are the eigenfunctions of the differential operator L. This eigenvalue problem is called the Hermite equation, although the term is also used for the closely related equation
whose solutions are the physicists' Hermite polynomials. With more general boundary conditions, the Hermite polynomials can be generalized to obtain more general analytic functions He(z) for a complex index. An explicit formula can be given in terms of a contour integral (Courant & Hilbert 1953).
Recursion relation
The sequence of Hermite polynomials also satisfies the recursion (probabilist) Individual coefficients are related by the following recursion formula:
and a[0,0]=1, a[1,0]=0, a[1,1]=1. (Assuming : ) (physicist) Individual coefficients are related by the following recursion formula:
and a[0,0]=1, a[1,0]=0, a[1,1]=2. The Hermite polynomials constitute an Appell sequence, i.e., they are a polynomial sequence satisfying the identity (probabilist) (physicist) or, equivalently, by Taylor expanding, (probabilist)
Hermite polynomials
31
(physicist) In consequence, for the m-th derivatives the following relations hold: (probabilist) (physicist) It follows that the Hermite polynomials also satisfy the recurrence relation (probabilist) (physicist) These last relations, together with the initial polynomials H0(x) and H1(x), can be used in practice to compute the polynomials quickly. Turn's inequalities are
Explicit expression
The physicists' Hermite polynomials can be written explicitly as
for odd values of n. These two equations may be combined into one using the floor function:
The probabilists' Hermite polynomials He have similar formulas, which may be obtained from these by replacing the power of 2x with the corresponding power of (2)x, and multiplying the entire sum by 2n/2.
Hermite polynomials
32
Generating function
The Hermite polynomials are given by the exponential generating function (probabilist) (physicist). This equality is valid for all x, t complex, and can be obtained by writing the Taylor expansion at x of the entire function z exp(z2) (in physicist's case). One can also derive the (physicist's) generating function by using Cauchy's Integral Formula to write the Hermite polynomials as
, one can evaluate the remaining integral using the calculus of residues and
Expected values
If X is a random variable with a normal distribution with standard deviation 1 and expected value then (probabilist) The moments of the standard normal may be read off directly from the relation for even indices
where
is the double factorial. Note that the above expression is a special case of the representation of
Asymptotic expansion
Asymptotically, as tends to infinity, the expansion (physicist[3]) holds true. For certain cases concerning a wider range of evaluation, it is necessary to include a factor for changing amplitude
This expansion is needed to resolve the wave-function of a quantum harmonic oscillator such that it agrees with the classical approximation in the limit of the correspondence principle. A finer approximation, which takes into account the uneven spacing of the zeros near the edges, makes use of the substitution , for , with which one has the uniform approximation
Hermite polynomials Similar approximations hold for the monotonic and transition regions. Specifically, if then for
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while for
with
Special Values
The Hermite polynomials evaluated at zero argument are called Hermite numbers.
(physicist)
Hermite polynomials
34
where D represents differentiation with respect to x, and the exponential is interpreted by expanding it as a power series. There are no delicate questions of convergence of this series when it operates on polynomials, since all but finitely many terms vanish. Since the power series coefficients of the exponential are well known, and higher order derivatives of the monomial xn can be written down explicitly, this differential operator representation gives rise to a concrete formula for the coefficients of Hn that can be used to quickly compute these polynomials. Since the formal expression for the Weierstrass transform W is eD2, we see that the Weierstrass transform of (2)nHen(x/2) is xn. Essentially the Weierstrass transform thus turns a series of Hermite polynomials into a corresponding Maclaurin series. The existence of some formal power series g(D) with nonzero constant coefficient, such that Hen(x) = g(D)xn, is another equivalent to the statement that these polynomials form an Appell sequencecf. W. Since they are an Appell sequence, they are a fortiori a Sheffer sequence.
Generalizations
The (probabilists') Hermite polynomials defined above are orthogonal with respect to the standard normal probability distribution, whose density function is
which has expected value 0 and variance 1. One may speak of Hermite polynomials
of variance , where is any positive number. These are orthogonal with respect to the normal probability distribution whose density function is
If
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is the umbral composition of the two polynomial sequences, and it can be shown to satisfy the identities
and
The last identity is expressed by saying that this parameterized family of polynomial sequences is a cross-sequence.
"Negative variance"
Since polynomial sequences form a group under the operation of umbral composition, one may denote by
the sequence that is inverse to the one similarly denoted but without the minus sign, and thus speak of Hermite polynomials of negative variance. For > 0, the coefficients of Hen[](x) are just the absolute values of the corresponding coefficients of Hen[](x). These arise as moments of normal probability distributions: The nth moment of the normal distribution with expected value and variance 2 is
where X is a random variable with the specified normal distribution. A special case of the cross-sequence identity then says that
Applications
Hermite functions
One can define the Hermite functions from the physicists' polynomials:
Since these functions contain the square root of the weight function, and have been scaled appropriately, they are orthonormal:
and form an orthonormal basis of L2(R). This fact is equivalent to the corresponding statement for Hermite polynomials (see above). The Hermite functions are closely related to the Whittaker function (Whittaker and Watson, 1962) :
and thereby to other parabolic cylinder functions. The Hermite functions satisfy the differential equation:
Hermite polynomials This equation is equivalent to the Schrdinger equation for a harmonic oscillator in quantum mechanics, so these functions are the eigenfunctions.
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Recursion relation
Following recursion relations of Hermite polynomials, the Hermite functions obey
as well as
Extending the first relation to the arbitrary m-th derivatives for any positive integer m leads to
This formula can be used in connection with the recurrence relations for Hen and the Hermite functions efficiently.
Hermite polynomials
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Cramr's inequality
The Hermite functions satisfy the following bound due to Harald Cramr
Choosing the unitary representation of the Fourier transform, the Fourier transform of the left hand side is given by
Equating like powers of t in the transformed versions of the left- and right-hand sides finally yields The Hermite functions n(x) are thus an orthonormal basis of L2(R) which diagonalizes the Fourier transform operator. In this case, we chose the unitary version of the Fourier transform, so the eigenvalues are(i)n. The ensuing resolution of the identity then serves to define powers, including fractional ones, of the Fourier transform, to wit a Fractional Fourier transform generalization.
Hermite polynomials
38
Completeness relation
The ChristoffelDarboux formula for Hermite polynomials reads
where is the Dirac delta function, (n) the Hermite functions, and (xy) represents the Lebesgue measure on the line y=x in R2, normalized so that its projection on the horizontal axis is the usual Lebesgue measure. This distributional identity follows by letting u1 in Mehler's formula, valid when 1<u<1:
The function (x,y) E(x,y;u) is the density for a Gaussian measure on R2 which is, when u is close to 1, very concentrated around the line y= x, and very spread out on that line. It follows that
when , g are continuous and compactly supported. This yields that can be expressed from the Hermite functions, as sum of a series of vectors in L2(R), namely
In order to prove the equality above for E(x,y;u), the Fourier transform of Gaussian functions will be used several times,
With this representation for Hn(x) and Hn(y), one sees that
and this implies the desired result, using again the Fourier transform of Gaussian kernels after performing the substitution
Hermite polynomials The proof of completeness by Mehler's formula is due to N.Wiener The Fourier integral and certain of its applications Cambridge Univ. Press 1933 reprinted Dover 1958
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Notes
[1] C. Hermite: Sur un nouveau dveloppement en srie de fonctions C. R Acad. Sci. Paris 58 1864 93-100; Oeuvres II 293-303 [2] P.L.Chebyshev: Sur le dveloppement des fonctions une seule variable Bull. Acad. Sci. St. Petersb. I 1859 193-200; Oeuvres I 501-508 [3] Abramowitz, p. 508-510 (http:/ / www. math. sfu. ca/ ~cbm/ aands/ page_508. htm), 13.6.38 and 13.5.16
References
Abramowitz, Milton; Stegun, Irene A., eds. (1965), "Chapter 22" (http://www.math.sfu.ca/~cbm/aands/ page_773.htm), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover, p.773, ISBN978-0486612720, MR 0167642 (http://www.ams.org/ mathscinet-getitem?mr=0167642). Courant, Richard; Hilbert, David (1953), Methods of Mathematical Physics, Volume I, Wiley-Interscience. Erdlyi, Arthur; Magnus, Wilhelm; Oberhettinger, Fritz; Tricomi, Francesco G. (1955), Higher transcendental functions. Vol. II, McGraw-Hill ( scan (http://www.nr.com/legacybooks)) Fedoryuk, M.V. (2001), "Hermite functions" (http://www.encyclopediaofmath.org/index.php?title=H/ h046980), in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN978-1-55608-010-4. Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, Ren F. (2010), "Orthogonal Polynomials" (http://dlmf.nist.gov/18), in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN978-0521192255, MR 2723248 (http://www.ams.org/mathscinet-getitem?mr=2723248) Laplace, P.S. (1810), Mm. Cl. Sci. Math. Phys. Inst. France 58: 279347 Suetin, P. K. (2001), "H/h047010" (http://www.encyclopediaofmath.org/index.php?title=H/h047010), in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN978-1-55608-010-4. Szeg, Gbor (1939, 1955), Orthogonal Polynomials, American Mathematical Society Wiener, Norbert (1958), The Fourier Integral and Certain of its Applications, New York: Dover Publications, ISBN0-486-60272-9 Whittaker, E. T.; Watson, G. N. (1962), 4th Edition, ed., A Course of Modern Analysis, London: Cambridge University Press Temme, Nico, Special Functions: An Introduction to the Classical Functions of Mathematical Physics, Wiley, New York, 1996
External links
Weisstein, Eric W., " Hermite Polynomial (http://mathworld.wolfram.com/HermitePolynomial.html)", MathWorld. Module for Hermite Polynomial Interpolation by John H. Mathews (http://math.fullerton.edu/mathews/n2003/ HermitePolyMod.html)
Jacobi polynomials
40
Jacobi polynomials
In mathematics, Jacobi polynomials (occasionally called hypergeometric polynomials) are a class of classical orthogonal polynomials. They are orthogonal with respect to the weight
on the interval [-1, 1]. The Gegenbauer polynomials, and thus also the Legendre, Zernike and Chebyshev polynomials, are special cases of the Jacobi polynomials.[1] The Jacobi polynomials were introduced by Carl Gustav Jacob Jacobi.
Definitions
Via the hypergeometric function
The Jacobi polynomials are defined via the hypergeometric function as follows:
where
is Pochhammer's symbol (for the rising factorial). In this case, the series for the hypergeometric
Rodrigues' formula
An equivalent definition is given by Rodrigues' formula:
In the special case that the four quantities n, n+, n+, and n++ are nonnegative integers, the Jacobi polynomial can be written as
(1)
The sum extends over all integer values of s for which the arguments of the factorials are nonnegative.
Jacobi polynomials
41
Basic properties
Orthogonality
The Jacobi polynomials satisfy the orthogonality condition
for , > 1. As defined, they are not orthonormal, the normalization being
Symmetry relation
The polynomials have the symmetry relation
Derivatives
The kth derivative of the explicit expression leads to
Differential equation
The Jacobi polynomial Pn(, ) is a solution of the second order linear homogeneous differential equation
Recurrence relation
The recurrent relation for the Jacobi polynomials is:
for n = 2, 3, ....
Generating function
The generating function of the Jacobi polynomials is given by
where
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42
and the "O" term is uniform on the interval [, -] for every > 0. The asymptotics of the Jacobi polynomials near the points 1 is given by the MehlerHeine formula
where the limits are uniform for z in a bounded domain. The asymptotics outside [1, 1] is less explicit.
Applications
Wigner d-matrix
The expression (1) allows the expression of the Wigner d-matrix djm,m() (for 0 4) in terms of Jacobi polynomials:
Notes
[1] The definition is in IV.1; the differential equation in IV.2; Rodrigues' formula is in IV.3; the generating function is in IV.4; the recurrent relation is in IV.5.
Further reading
Andrews, George E.; Askey, Richard; Roy, Ranjan (1999), Special functions, Encyclopedia of Mathematics and its Applications 71, Cambridge University Press, ISBN978-0-521-62321-6; 978-0-521-78988-2 Check |isbn= value (help), MR 1688958 (http://www.ams.org/mathscinet-getitem?mr=1688958) Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, Ren F. (2010), "Orthogonal Polynomials" (http://dlmf.nist.gov/18), in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN978-0521192255, MR 2723248 (http://www.ams.org/mathscinet-getitem?mr=2723248)
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43
External links
Weisstein, Eric W., " Jacobi Polynomial (http://mathworld.wolfram.com/JacobiPolynomial.html)", MathWorld.
Legendre polynomials
In mathematics, Legendre functions are solutions to Legendre's differential equation:
They are named after Adrien-Marie Legendre. This ordinary differential equation is frequently encountered in physics and other technical fields. In particular, it occurs when solving Laplace's equation (and related partial differential equations) in spherical coordinates. The Legendre differential equation may be solved using the standard power series method. The equation has regular singular points at x=1 so, in general, a series solution about the origin will only converge for |x|<1. When n is an integer, the solution Pn(x) that is regular at x=1 is also regular at x=1, and the series for this solution terminates (i.e. it is a polynomial). These solutions for n=0,1,2,... (with the normalization Pn(1)=1) form a polynomial sequence of orthogonal polynomials called the Legendre polynomials. Each Legendre polynomial Pn(x) is an nth-degree polynomial. It may be expressed using Rodrigues' formula:
That these polynomials satisfy the Legendre differential equation (1) follows by differentiating (n+1) times both sides of the identity
and employing the general Leibniz rule for repeated differentiation. The Pn can also be defined as the coefficients in a Taylor series expansion:
Recursive definition
Expanding the Taylor series in equation (1) for the first two terms gives
for the first two Legendre Polynomials. To obtain further terms without resorting to direct expansion of the Taylor series, equation (1) is differentiated with respect to t on both sides and rearranged to obtain
Replacing the quotient of the square root with its definition in (1), and equating the coefficients of powers of t in the resulting expansion gives Bonnets recursion formula
This relation, along with the first two polynomials P0 and P1, allows the Legendre Polynomials to be generated recursively.
44
where the latter, which is immediate from the recursion formula, expresses the Legendre polynomials by simple monomials and involves the multiplicative formula of the binomial coefficient. The first few Legendre polynomials are:
n 0 1 2 3 4 5 6 7 8 9 10
Legendre polynomials
45
Orthogonality
An important property of the Legendre polynomials is that they are orthogonal with respect to the L2 inner product on the interval 1x1:
(where mn denotes the Kronecker delta, equal to 1 if m=n and to 0 otherwise). In fact, an alternative derivation of the Legendre polynomials is by carrying out the Gram-Schmidt process on the polynomials {1,x,x2,...} with respect to this inner product. The reason for this orthogonality property is that the Legendre differential equation can be viewed as a SturmLiouville problem, where the Legendre polynomials are eigenfunctions of a Hermitian differential operator:
where
and
and
respectively and
the Coulomb potential associated to a point charge. The expansion using Legendre polynomials might be useful, for instance, when integrating this expression over a continuous mass or charge distribution. Legendre polynomials occur in the solution of Laplace equation of the potential, , in a charge-free
region of space, using the method of separation of variables, where the boundary conditions have axial symmetry (no dependence on an azimuthal angle). Where is the axis of symmetry and is the angle between the position of the observer and the axis (the zenith angle), the solution for the potential will be
and
They also appear when solving Schrdinger equation in three dimensions for a central force. Legendre polynomials in multipole expansions
Legendre polynomials
46
Legendre polynomials are also useful in expanding functions of the form (this is the same as before, written a little differently):
which arise naturally in multipole expansions. The left-hand side of the equation is the generating function for the Legendre polynomials. As an example, the electric potential (Figure2) varies like
Figure 2
(in spherical
If the radius r of the observation point P is greater than a, the potential may be expanded in the Legendre polynomials
where we have defined =a/r<1 and x=cos. This expansion is used to develop the normal multipole expansion. Conversely, if the radius r of the observation point P is smaller than a, the potential may still be expanded in the Legendre polynomials as above, but with a and r exchanged. This expansion is the basis of interior multipole expansion.
Since the differential equation and the orthogonality property are independent of scaling, the Legendre polynomials' definitions are "standardized" (sometimes called "normalization", but note that the actual norm is not unity) by being scaled so that
As discussed above, the Legendre polynomials obey the three term recurrence relation known as Bonnets recursion formula
and
or equivalently
Legendre polynomials
47
where
From Bonnets recursion formula one obtains by induction the explicit representation
Legendre polynomials
48
Legendre functions
As well as polynomial solutions, the Legendre equation has non-polynomial solutions represented by infinite series. These are the Legendre functions of the second kind, denoted by .
Notes
[1] M. Le Gendre, "Recherches sur l'attraction des sphrodes homognes," Mmoires de Mathmatiques et de Physique, prsents l'Acadmie Royale des Sciences, par divers savans, et lus dans ses Assembles, Tome X, pp. 411-435 (Paris, 1785). [Note: Legendre submitted his findings to the Academy in 1782, but they were published in 1785.] Available on-line (in French) at: http:/ / edocs. ub. uni-frankfurt. de/ volltexte/ 2007/ 3757/ pdf/ A009566090. pdf . [2] Jackson, J.D. Classical Electrodynamics, 3rd edition, Wiley & Sons, 1999. page 103
References
Abramowitz, Milton; Stegun, Irene A., eds. (1965), "Chapter 8" (http://www.math.sfu.ca/~cbm/aands/ page_332.htm), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover, p.332, ISBN978-0486612720, MR 0167642 (http://www.ams.org/ mathscinet-getitem?mr=0167642) See also chapter 22 (http://www.math.sfu.ca/~cbm/aands/page_773.htm). Bayin, S.S. (2006), Mathematical Methods in Science and Engineering, Wiley, Chapter 2. Belousov, S. L. (1962), Tables of normalized associated Legendre polynomials, Mathematical tables 18, Pergamon Press. Courant, Richard; Hilbert, David (1953), Methods of Mathematical Physics, Volume 1, New York: Interscience Publischer, Inc. Dunster, T. M. (2010), "Legendre and Related Functions" (http://dlmf.nist.gov/14), in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN978-0521192255, MR 2723248 (http://www.ams.org/ mathscinet-getitem?mr=2723248) Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, Ren F. (2010), "Orthogonal Polynomials" (http://dlmf.nist.gov/18), in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN978-0521192255, MR 2723248 (http://www.ams.org/mathscinet-getitem?mr=2723248) Refaat El Attar (2009), Legendre Polynomials and Functions, CreateSpace, ISBN978-1-4414-9012-4
Legendre polynomials
49
External links
A quick informal derivation of the Legendre polynomial in the context of the quantum mechanics of hydrogen (http://www.physics.drexel.edu/~tim/open/hydrofin) Hazewinkel, Michiel, ed. (2001), "Legendre polynomials" (http://www.encyclopediaofmath.org/index. php?title=p/l058050), Encyclopedia of Mathematics, Springer, ISBN978-1-55608-010-4 Wolfram MathWorld entry on Legendre polynomials (http://mathworld.wolfram.com/LegendrePolynomial. html) Module for Legendre Polynomials by John H. Mathews (http://math.fullerton.edu/mathews/n2003/ LegendrePolyMod.html) Dr James B. Calvert's article on Legendre polynomials from his personal collection of mathematics (http://www. du.edu/~jcalvert/math/legendre.htm) The Legendre Polynomials by Carlyle E. Moore (http://www.morehouse.edu/facstaff/cmoore/Legendre Polynomials.htm) Legendre Polynomials from Hyperphysics (http://hyperphysics.phy-astr.gsu.edu/hbase/math/legend.html)
Chebyshev polynomials
In mathematics the Chebyshev polynomials, named after Pafnuty Chebyshev,[1] are a sequence of orthogonal polynomials which are related to de Moivre's formula and which can be defined recursively. One usually distinguishes between Chebyshev polynomials of the first kind which are denoted Tn and Chebyshev polynomials of the second kind which are denoted Un. The letter T is used because of the alternative transliterations of the name Chebyshev as Tchebycheff, Tchebyshev (French) or Tschebyschow (German). The Chebyshev polynomials Tn or Un are polynomials of degree n and the sequence of Chebyshev polynomials of either kind composes a polynomial sequence. Chebyshev polynomials are polynomials with the largest possible leading coefficient, but subject to the condition that their absolute value is bounded on the interval by 1. They are also the extremal polynomials for many other properties.[2] Chebyshev polynomials are important in approximation theory because the roots of the Chebyshev polynomials of the first kind, which are also called Chebyshev nodes, are used as nodes in polynomial interpolation. The resulting interpolation polynomial minimizes the problem of Runge's phenomenon and provides an approximation that is close to the polynomial of best approximation to a continuous function under the maximum norm. This approximation leads directly to the method of ClenshawCurtis quadrature. In the study of differential equations they arise as the solution to the Chebyshev differential equations
and
for the polynomials of the first and second kind, respectively. These equations are special cases of the SturmLiouville differential equation.
Chebyshev polynomials
50
Definition
The Chebyshev polynomials of the first kind are defined by the recurrence relation
The generating function relevant for 2-dimensional potential theory and multipole expansion is
The Chebyshev polynomials of the second kind are defined by the recurrence relation
Trigonometric definition
The Chebyshev polynomials of the first kind can be defined as the unique polynomials satisfying
for n = 0, 1, 2, 3, ... which is a variant (equivalent transpose) of Schrder's equation, viz. Tn(x) is functionally conjugate to nx, codified in the nesting property below. Further compare to the spread polynomials, in the section below. The polynomials of the second kind satisfy:
That cos(nx) is an nth-degree polynomial in cos(x) can be seen by observing that cos(nx) is the real part of one side of de Moivre's formula, and the real part of the other side is a polynomial in cos(x) and sin(x), in which all powers of sin(x) are even and thus replaceable through the identity cos2(x) + sin2(x) = 1. This identity is quite useful in conjunction with the recursive generating formula, inasmuch as it enables one to calculate the cosine of any integral multiple of an angle solely in terms of the cosine of the base angle. Evaluating the first two Chebyshev polynomials,
Chebyshev polynomials
51
and
and so forth. Two immediate corollaries are the composition identity (or nesting property specifying a semigroup)
, where n is odd.
, where n is even. The recurrence relationship of the derivative of Chebyshev polynomials can be derived from these relations
This relationship is used in the Chebyshev spectral method of solving differential equations. Equivalently, the two sequences can also be defined from a pair of mutual recurrence equations:
Chebyshev polynomials
52
, then
Note that both these equations and the trigonometric equations take a simpler form if we, like some works, follow the alternate convention of denoting our Un (the polynomial of degree n) with Un+1 instead. Turn's inequalities for the Chebyshev polynomials are and
Explicit expressions
Different approaches to defining Chebyshev polynomials lead to different explicit expressions such as:
Chebyshev polynomials
53
where
is a hypergeometric function.
Properties
Roots and extrema
A Chebyshev polynomial of either kind with degree n has n different simple roots, called Chebyshev roots, in the interval [1,1]. The roots of the Chebyshev polynomial of the first kind are sometimes called Chebyshev nodes because they are used as nodes in polynomial interpolation. Using the trigonometric definition and the fact that
The extrema of Tn on the interval 1 x 1 are located at One unique property of the Chebyshev polynomials of the first kind is that on the interval 1 x 1 all of the extrema have values that are either 1 or 1. Thus these polynomials have only two finite critical values, the defining property of Shabat polynomials. Both the first and second kinds of Chebyshev polynomial have extrema at the endpoints, given by:
Chebyshev polynomials
54
The last two formulas can be numerically troublesome due to the division by zero (0/0 indeterminate form, specifically) at x = 1 and x = 1. It can be shown that:
Proof The second derivative of the Chebyshev polynomial of the first kind is
which, if evaluated as shown above, poses a problem because it is indeterminate at x = 1. Since the function is a polynomial, (all of) the derivatives must exist for all real numbers, so the taking to limit on the expression above should yield the desired value:
where only
Since the limit as a whole must exist, the limit of the numerator and denominator must independently exist, and
The denominator (still) limits to zero, which implies that the numerator must be limiting to zero, i.e. which will be useful later on. Since the numerator and denominator are both limiting to zero, L'Hpital's rule applies:
Chebyshev polynomials
55
being important.
This latter result is of great use in the numerical solution of eigenvalue problems. Concerning integration, the first derivative of the Tn implies that
and the recurrence relation for the first kind polynomials involving derivatives establishes that
Orthogonality
Both the Tn and the Un form a sequence of orthogonal polynomials. The polynomials of the first kind are orthogonal with respect to the weight
This can be proven by letting x = cos () and using the defining identity Tn(cos ()) = cos (n). Similarly, the polynomials of the second kind are orthogonal with respect to the weight
56
Minimal -norm
For any given n 1, among the polynomials of degree n with leading coefficient 1,
is the one of which the maximal absolute value on the interval [1,1] is minimal. This maximal absolute value is
Proof Let's assume that interval [1,1] less than Define is a polynomial of degree n with leading coefficient 1 with maximal absolute value on the .
we have
is a
polynomial of degree n 1, so the fundamental theorem of algebra implies it has at most n 1 roots.
Chebyshev polynomials
57
Other properties
The Chebyshev polynomials are a special case of the ultraspherical or Gegenbauer polynomials, which themselves are a special case of the Jacobi polynomials: For every nonnegative integer n, Tn(x) and Un(x) are both polynomials of degree n. They are even or odd functions of x as n is even or odd, so when written as polynomials of x, it only has even or odd degree terms respectively. In fact,
and
The leading coefficient of Tn is 2n 1 if 1 n, but 1 if 0 = n. Tn are a special case of Lissajous curves with frequency ratio equal to n. Several polynomial sequences like Lucas polynomials (Ln), Dickson polynomials(Dn), Fibonacci polynomials(Fn) are related to Chebyshev polynomials Tn and Un. The Chebyshev polynomials of the first kind satisfy the relation
which is easily proved from the product-to-sum formula for the cosine. The polynomials of the second kind satisfy the similar relation
we have the analogous formula . For , and , which follows from the fact that this holds by definition for Let . Then and are commuting polynomials: , as is evident in the Abelian nesting property specified above. .
Chebyshev polynomials
58
Examples
The first few Chebyshev polynomials of the first kind are A028297
The first few Chebyshev polynomials of the first kind in the domain 1 < x < 1: The flat T0, T1, T2, T3, T4 and T5.
The first few Chebyshev polynomials of the second kind in the domain 1<x<1: The flat U0, U1, U2, U3, U4 and U5. Although not visible in the image, Un(1) = n + 1 and Un(1) = (n + 1)(1)n.
Chebyshev polynomials
59
As a basis set
In the appropriate Sobolev space, the set of Chebyshev polynomials form a orthonormal basis, so that a function in the same space can, on 1 x 1 be expressed via the expansion:
Furthermore, as mentioned previously, the Chebyshev polynomials form an orthogonal basis which (among other things) implies that the coefficients an can be determined easily through the application of an inner product. This sum is called a Chebyshev series or a Chebyshev expansion. Since a Chebyshev series is related to a Fourier cosine series through a change of variables, all of the theorems, identities, etc. that apply to Fourier series have a Chebyshev counterpart. These attributes include: The Chebyshev polynomials form a complete orthogonal system. The Chebyshev series converges to (x) if the function is piecewise smooth and continuous. The smoothness requirement can be relaxed in most cases as long as there are a finite number of discontinuities in (x) and its derivatives. At a discontinuity, the series will converge to the average of the right and left limits.
The non-smooth function (top) y = x3H(x), where H is the Heaviside step function, and (bottom) the 5th partial sum of its Chebyshev expansion. The 7th sum is indistinguishable from the original function at the resolution of the graph.
The abundance of the theorems and identities inherited from Fourier series make the Chebyshev polynomials important tools in numeric analysis; for example they are the most popular general purpose basis functions used in the spectral method, often in favor of trigonometric series due to generally faster convergence for continuous functions (Gibbs' phenomenon is still a problem).
Example 1
Consider the Chebyshev expansion of . One can express
One can find the coefficients either through the application of an inner product or by the discrete orthogonality condition. For the inner product,
which gives
Alternatively, when you cannot evaluate the inner product of the function you are trying to approximate, the discrete orthogonality condition gives
Chebyshev polynomials
60
where
Example 2
To provide another example:
Partial sums
The partial sums of
are very useful in the approximation of various functions and in the solution of differential equations (see spectral method). Two common methods for determining the coefficients an are through the use of the inner product as in Galerkin's method and through the use of collocation which is related to interpolation. As an interpolant, the N coefficients of the (N 1)th partial sum are usually obtained on the ChebyshevGaussLobatto[4] points (or Lobatto grid), which results in minimum error and avoids Runge's phenomenon associated with a uniform grid. This collection of points corresponds to the extrema of the highest order polynomial in the sum, plus the endpoints and is given by:
Chebyshev polynomials
61
Spread polynomials
The spread polynomials are in a sense equivalent to the Chebyshev polynomials of the first kind, but enable one to avoid square roots and conventional trigonometric functions in certain contexts, notably in rational trigonometry.
Notes
[1] Chebyshev polynomials were first presented in: P. L. Chebyshev (1854) "Thorie des mcanismes connus sous le nom de paralllogrammes," Mmoires des Savants trangers prsents lAcadmie de Saint-Ptersbourg, vol. 7, pages 539586. [2] Rivlin, Theodore J. The Chebyshev polynomials.Pure and Applied Mathematics.Wiley-Interscience [John Wiley & Sons], New York-London-Sydney,1974. Chapter 2, "Extremal Properties", pp. 56--123. [3] Jeroen Demeyer Diophantine Sets over Polynomial Rings and Hilbert's Tenth Problem for Function Fields (http:/ / cage. ugent. be/ ~jdemeyer/ phd. pdf), Ph.D. theses (2007), p.70. [4] Chebyshev Interpolation: An Interactive Tour (http:/ / www. joma. org/ images/ upload_library/ 4/ vol6/ Sarra/ Chebyshev. html)
References
Abramowitz, Milton; Stegun, Irene A., eds. (1965), "Chapter 22" (http://www.math.sfu.ca/~cbm/aands/ page_773.htm), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover, p.773, ISBN978-0486612720, MR 0167642 (http://www.ams.org/ mathscinet-getitem?mr=0167642). Dette, Holger (1995), A Note on Some Peculiar Nonlinear Extremal Phenomena of the Chebyshev Polynomials (http://journals.cambridge.org/download.php?file=/PEM/PEM2_38_02/S001309150001912Xa.pdf& code=c12b9a2fc1aba9e27005a5003ed21b36), Proceedings of the Edinburgh Mathematical Society 38, 343-355 Eremenko, A.; Lempert, L. (1994), An Extremal Problem For Polynomials (http://www.math.purdue.edu/ ~eremenko/dvi/lempert.pdf), Proceedings of the American Mathematical Society, Volume 122, Number 1, 191-193 Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, Ren F. (2010), "Orthogonal Polynomials" (http://dlmf.nist.gov/18), in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN978-0521192255, MR 2723248 (http://www.ams.org/mathscinet-getitem?mr=2723248) Remes, Eugene, On an Extremal Property of Chebyshev Polynomials (http://www.math.technion.ac.il/hat/ fpapers/remeztrans.pdf) Suetin, P.K. (2001), "C/c021940" (http://www.encyclopediaofmath.org/index.php?title=C/c021940), in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN978-1-55608-010-4
External links
Weisstein, Eric W., " Chebyshev Polynomial of the First Kind (http://mathworld.wolfram.com/ ChebyshevPolynomialoftheFirstKind.html)", MathWorld. Module for Chebyshev Polynomials by John H. Mathews (http://math.fullerton.edu/mathews/n2003/ ChebyshevPolyMod.html) Chebyshev Interpolation: An Interactive Tour (http://www.joma.org/images/upload_library/4/vol6/Sarra/ Chebyshev.html), includes illustrative Java applet. Numerical Computing with Functions: The Chebfun Project (http://www.maths.ox.ac.uk/chebfun/) Is there an intuitive explanation for an extremal property of Chebyshev polynomials? (http://mathoverflow.net/ questions/25534/is-there-an-intuitive-explanation-for-an-extremal-property-of-chebyshev-polynomia)
Gegenbauer polynomials
62
Gegenbauer polynomials
In mathematics, Gegenbauer polynomials or ultraspherical polynomials C() 2 1/2 n(x) are orthogonal polynomials on the interval [1,1] with respect to the weight function (1x ) . They generalize Legendre polynomials and Chebyshev polynomials, and are special cases of Jacobi polynomials. They are named after Leopold Gegenbauer.
Characterizations
A variety of characterizations of the Gegenbauer polynomials are available. The polynomials can be defined in terms of their generating function (Stein & Weiss 1971, IV.2):
Gegenbauer polynomials are particular solutions of the Gegenbauer differential equation (Suetin 2001):
When =1/2, the equation reduces to the Legendre equation, and the Gegenbauer polynomials reduce to the Legendre polynomials. They are given as Gaussian hypergeometric series in certain cases where the series is in fact finite:
(Abramowitz & Stegun p.561 [1]). Here (2)n is the rising factorial. Explicitly,
in which
Gegenbauer polynomials
63
Applications
The Gegenbauer polynomials appear naturally as extensions of Legendre polynomials in the context of potential theory and harmonic analysis. The Newtonian potential in Rn has the expansion, valid with =(n2)/2,
When n=3, this gives the Legendre polynomial expansion of the gravitational potential. Similar expressions are available for the expansion of the Poisson kernel in a ball (Stein & Weiss 1971). It follows that the quantities are spherical harmonics, when regarded as a function of x only.
They are, in fact, exactly the zonal spherical harmonics, up to a normalizing constant. Gegenbauer polynomials also appear in the theory of Positive-definite functions. The AskeyGasper inequality reads
References
Abramowitz, Milton; Stegun, Irene A., eds. (1965), "Chapter 22" [3], Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover, p.773, ISBN978-0486612720, MR0167642 [4] . Bayin, S.S. (2006), Mathematical Methods in Science and Engineering, Wiley, Chapter 5. Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, Ren F. (2010), "Orthogonal Polynomials" [5], in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN978-0521192255, MR2723248 [6] Stein, Elias; Weiss, Guido (1971), Introduction to Fourier Analysis on Euclidean Spaces, Princeton, N.J.: Princeton University Press, ISBN978-0-691-08078-9. Suetin, P.K. (2001), "Ultraspherical polynomials" [7], in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN978-1-55608-010-4.
Gegenbauer polynomials
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References
[1] [2] [3] [4] [5] [6] [7] http:/ / www. math. sfu. ca/ ~cbm/ aands/ page_561. htm http:/ / www. math. sfu. ca/ ~cbm/ aands/ page_774. htm http:/ / www. math. sfu. ca/ ~cbm/ aands/ page_773. htm http:/ / www. ams. org/ mathscinet-getitem?mr=0167642 http:/ / dlmf. nist. gov/ 18 http:/ / www. ams. org/ mathscinet-getitem?mr=2723248 http:/ / www. encyclopediaofmath. org/ index. php?title=U/ u095030
Laguerre polynomials
In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834 1886), are solutions of Laguerre's equation:
which is a second-order linear differential equation. This equation has nonsingular solutions only if n is a non-negative integer. The associated Laguerre polynomials (also named Sonin polynomials after Nikolay Yakovlevich Sonin in some older books) are solutions of
The Laguerre polynomials are also used for Gaussian quadrature to numerically compute integrals of the form
These polynomials, usually denoted L0,L1,..., are a polynomial sequence which may be defined by the Rodrigues formula,
reducing to the closed form of a following section. They are orthogonal polynomials with respect to an inner product
The sequence of Laguerre polynomials n! Ln is a Sheffer sequence, ddx Ln = (ddx1) Ln1. The Rook polynomials in combinatorics are more or less the same as Laguerre polynomials, up to elementary changes of variables. The Laguerre polynomials arise in quantum mechanics, in the radial part of the solution of the Schrdinger equation for a one-electron atom. They also describe the static Wigner functions of oscillator systems in quantum mechanics in phase space. They further enter in the quantum mechanics of the 3D isotropic harmonic oscillator. Physicists sometimes use a definition for the Laguerre polynomials which is larger by a factor of n! than the definition used here. (Likewise, some physicist may use somewhat different definitions of the so-called associated Laguerre polynomials.)
Laguerre polynomials
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and then using the following recurrence relation for any k1:
Laguerre polynomials The exponential generating function for them likewise follows,
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are called generalized Laguerre polynomials, or associated Laguerre polynomials. The simple Laguerre polynomials are included in the associated polynomials, through = 0,
When n is an integer the function reduces to a polynomial of degreen. It has the alternative expression[3]
in terms of Kummer's function of the second kind. The closed form for these associated Laguerre polynomials of degree n is[4]
derived by applying Leibniz's theorem for differentiation of a product to Rodrigues' formula. The first few generalized Laguerre polynomials are:
The coefficient of the leading term is (1)n/n!; The constant term, which is the value at0, is
The polynomials' asymptotic behaviour for large n, but fixed and x>0, is given by[5][6]
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and summarizing by
where
Recurrence relations
The addition formula for Laguerre polynomials:[7] . Laguerre's polynomials satisfy the recurrence relations
in particular
and
or
moreover
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This points to a special case (=0) of the formula above: for integer =k the generalized polynomial may be written notation for a derivative. Moreover, this following equation holds , the shift by k sometimes causing confusion with the usual parenthesis
The derivate with respect to the second variable has the surprising form
Laguerre polynomials which may be compared with the equation obeyed by the kth derivative of the ordinary Laguerre polynomial,
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where
Orthogonality
The associated Laguerre polynomials are orthogonal over [0,) with respect to the measure with weighting function xex:[8]
If
denoted the Gamma distribution then the orthogonality relation can be written as
The associated, symmetric kernel polynomial has the representations (ChristoffelDarboux formula)[citation needed]
recursively
Moreover, in the associated L2[0,)-space. Turn's inequalities can be derived here, which is
The following integral is needed in the quantum mechanical treatment of the hydrogen atom,
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Series expansions
Let a function have the (formal) series expansion
Then
, iff
for the exponential function. The incomplete gamma function has the representation
Multiplication theorems
Erdlyi gives the following two multiplication theorems [9]
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As a contour integral
Given the generating function specified above, the polynomials may be expressed in terms of a contour integral
and where the Hn(x) are the Hermite polynomials based on the weighting function exp(x2), the so-called "physicist's version." Because of this, the generalized Laguerre polynomials arise in the treatment of the quantum harmonic oscillator.
where
is the Pochhammer symbol (which in this case represents the rising factorial).
Poisson Kernel
Notes
[1] [2] [3] [4] [5] [6] A&S p. 781 A&S p.509 A&S p.510 A&S p. 775 G. Szeg, "Orthogonal polynomials", 4th edition, Amer. Math. Soc. Colloq. Publ., vol. 23, Amer. Math. Soc., Providence, RI, 1975, p. 198. D. Borwein, J. M. Borwein, R. E. Crandall, "Effective Laguerre asymptotics", SIAM J. Numer. Anal., vol. 46 (2008), no. 6, pp. 3285-3312, http:/ / dx. doi. org/ 10. 1137/ 07068031X [7] A&S equation (22.12.6), p. 785 [8] A&S p. 774 [9] C. Truesdell, " On the Addition and Multiplication Theorems for the Special Functions (http:/ / www. pnas. org/ cgi/ reprint/ 36/ 12/ 752. pdf)", Proceedings of the National Academy of Sciences, Mathematics, (1950) pp.752-757.
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References
Abramowitz, Milton; Stegun, Irene A., eds. (1965), "Chapter 22" (http://www.math.sfu.ca/~cbm/aands/ page_773.htm), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover, p.773, ISBN978-0486612720, MR 0167642 (http://www.ams.org/ mathscinet-getitem?mr=0167642). Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, Ren F. (2010), "Orthogonal Polynomials" (http://dlmf.nist.gov/18), in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN978-0521192255, MR 2723248 (http://www.ams.org/mathscinet-getitem?mr=2723248) B. Spain, M.G. Smith, Functions of mathematical physics, Van Nostrand Reinhold Company, London, 1970. Chapter 10 deals with Laguerre polynomials. Hazewinkel, Michiel, ed. (2001), "Laguerre polynomials" (http://www.encyclopediaofmath.org/index. php?title=p/l057310), Encyclopedia of Mathematics, Springer, ISBN978-1-55608-010-4 Eric W. Weisstein, " Laguerre Polynomial (http://mathworld.wolfram.com/LaguerrePolynomial.html)", From MathWorldA Wolfram Web Resource. George Arfken and Hans Weber (2000). Mathematical Methods for Physicists. Academic Press. ISBN0-12-059825-6. S. S. Bayin (2006), Mathematical Methods in Science and Engineering, Wiley, Chapter 3.
External links
Timothy Jones. "The Legendre and Laguerre Polynomials and the elementary quantum mechanical model of the Hydrogen Atom" (http://www.physics.drexel.edu/~tim/open/hydrofin). Weisstein, Eric W., " Laguerre polynomial (http://mathworld.wolfram.com/LaguerrePolynomial.html)", MathWorld.
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Eigenfunction
In mathematics, an eigenfunction of a linear operator, A, defined on some function space, is any non-zero function f in that space that returns from the operator exactly as is, except for a multiplicative scaling factor. More precisely, one has
for some scalar, , the corresponding eigenvalue. The solution of the differential eigenvalue problem also depends on any boundary conditions required of . In each case there are only certain eigenvalues solution for For example, ( (with each ) that admit a corresponding belonging to the eigenvalue .
This solution of the vibrating drum problem is, at any point in time, an eigenfunction of the Laplace operator on a disk.
. If boundary conditions are applied to this satisfy the boundary which when input . .
[1]
conditions, generating corresponding discrete eigenvalues into the system, produces a response
Specifically, in the study of signals and systems, the eigenfunction of a system is the signal with the complex constant
Examples
Derivative operator
A widely used class of linear operators acting on function spaces are the differential operators on function spaces. As an example, on the space of infinitely differentiable real functions of a real argument , the process of differentiation is a linear operator since
and
in
and in
The functions that satisfy this equation are commonly called eigenfunctions. For the derivative operator eigenfunction is a function that, when differentiated, yields a constant times the original function. That is,
, an
for all
The derivative operator is defined also for complex-valued functions of a complex argument. In the complex version of the space , the eigenvalue equation has a solution for any complex constant . The spectrum of the operator is therefore the whole complex plane. This is an example of a continuous spectrum.
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Applications
Vibrating strings
Let denote the sideways displacement of a stressed elastic chord, such as the vibrating strings of a string instrument, as a function of the position along the string and of time . From the laws of mechanics, applied to infinitesimal portions of the string, one can deduce that the function satisfies the partial differential equation
The shape of a standing wave in a string fixed at its boundaries is an example of an eigenfunction of a differential operator. The admissible eigenvalues are governed by the length of the string and determine the frequency of oscillation.
which is called the (one-dimensional) wave equation. Here is a constant that depends on the tension and mass of the string. This problem is amenable to the method of separation of variables. If we assume that product of the form and Each of these is an eigenvalue equation, for eigenvalues and , the equations are satisfied by the functions and where with and are arbitrary real constants. If we impose boundary conditions (that the ends of the string are fixed at and , for example) we can constrain the eigenvalues. For those boundary and , respectively. For any values of , we can form a pair of ordinary differential equations: can be written as the
, where
clamped string supports a family of standing waves of the form From the point of view of our musical instrument, the frequency called the th overtone. is the frequency of the th harmonic, which is
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Quantum mechanics
Eigenfunctions play an important role in many branches of physics. An important example is quantum mechanics, where the Schrdinger equation , with
where
with eigenvalues
satisfy Schrdinger's equation leads to a natural basis for quantum mechanics and an allowable energy state of the system. The success of this
equation in explaining the spectral characteristics of hydrogen is considered one of the greatest triumphs of 20th century physics. Due to the nature of Hermitian Operators such as the Hamiltonian operator , its eigenfunctions are orthogonal functions. This is not necessarily the case for eigenfunctions of other operators (such as the example above). Orthogonal functions , have the property that mentioned
where whenever
is the complex conjugate of , in which case the set is said to be orthogonal. Also, it is linearly independent.
Notes
[1] Bernd Girod, Rudolf Rabenstein, Alexander Stenger, Signals and systems, 2nd ed., Wiley, 2001, ISBN 0-471-98800-6 p. 49
References
Methods of Mathematical Physics by R. Courant, D. Hilbert ISBN 0-471-50447-5 (Volume 1 Paperback) ISBN 0-471-50439-4 (Volume 2 Paperback) ISBN 0-471-17990-6 (Hardback)
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License
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License
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