OFFSET
0,2
COMMENTS
See A199577 for general comments on associated Laguerre-Sonin(e) polynomials of order m, and the Ismail reference.
The monic row polynomials are La_n(1;1,x) = sum(a(n,k)*x^k,k=0..n), with the three term recurrence
La_n(1;1,x) = (x-2*(n+1))*La_{n-1}{1;1,x) - n*(n+1)*La_{n-2}{1;1,x), La_{-1}{1;1,x)=0, La_0(1;1,x)=1.
In the Ismail reference the non-monic associated Laguerre polynomials of order 1 appear on p. 160 in Theorem 5.6.1, eq. 5.6.11. The connection is: La_n(1;1,x)= L_n^{(alpha=1)}(x;1)*(n+1)!*(-1)^n.
The e.g.f. gLa(1;1,z,x) for La_n(1;1,x) can be obtained from the o.g.f. G(1;1,z,x) for the non-monic version L_n^{alpha=1}(x;1) by gLa(1;1,z,x)= (d/dz) (z*G(1;1,-z,x)).
G(1;1,z,x) satisfies an ordinary first order inhomogeneous differential equation:
(d/dz) G(1;1,z,x) = (3/(1-z)+(1-x)/(1-z)^2-1/(z*(1-z)^2 z)^2))* G(1;1,z,x) + 1/(z*(1-z)^2), with G(1;1,z=0,x)=1. The standard solution is:
G(1;1,z,x) = (exp(-x*z/(1-z))-1+z-x*exp(-x/(1-z))* (Ei(1,-x/(1-z))-Ei(1,-x)))/(z*(1-z)^2), with the exponential integral Ei(1,y)=int(exp(-t)/t,t=y..infty).
REFERENCES
M. E. H. Ismail (two chapters by W. Van Assche), Classical and Quantum Orthogonal Polynomials in One Variable, Cambridge University Press, 2005.
FORMULA
a(n,k)=[x^k] La_n(1;1,x), n>=0, k=0,...,n.
a(n,k)= (-1)^(n-k)*((n+1)*(n+1)!/((k+1)*(k+1)!))*
binomial(n,k)*hypergeom(-(n-k),k,1; -(n+1),k+2; 1), 0<=k<=n. [Ismail, p. 161, eq. (5.6.18)].
The e.g.f. gLa(1;1,z,x) for La_n(1;1,x) is exp(-x/(1+z))*x*(x-2*(1+z))*(Ei(1,-x/(1+z)) - Ei(1,-x))/(1+z)^4 + exp(x*z/(1+z))*(-x+2*(1+z))/(1+z)^4 +(1+z+x)/(1+z)^3 -2/(1+z)^2, with the exponential integral Ei.
The e.g.f. gLa(1;1,z,x) for the Euler-derivative
x*(d/dx) La_n(1;1,x) is x*exp(-x/(1+z))*(2*(1+z)-x)*
(Ei(1,-x/(1+z)) - Ei(1,-x))/(1+z)^4 + (1+z-x)*(1-exp(x*z/(1+z)))/(1+z)^3.
From this follows La_n(1;1,x) = (n+1)*La_n(1,x) -
x*(d/dx)La_n(1;0,x). For La_n(1;0,x) see A199577 where it is called La_n(1;x).
EXAMPLE
The triangle begins:
n\k 0 1 2 3 4 5 6 7 ...
0: 1
1: -4 1
2: 18 -10 1
3: -96 86 -18 1
4: 600 -756 246 -28 1
5: -4320 7092 -3168 552 -40 1
6: 35280 -71856 41112 -9720 1070 -54 1
7:-322560 787824 -552240 165720 -24600 1878 -70 1
...
MAPLE
La := proc(n, x)
option remember;
if n= -1 then
0;
elif n = 0 then
1;
else
(x-2*n-2)*procname(n-1, x)-n*(n+1)*procname(n-2, x) ;
end if;
end proc:
A201201 := proc(n, k)
coeftayl( La(n, x), x=0, k) ;
end proc:
seq(seq(A201201(n, k), k=0..n), n=0..12) ; # R. J. Mathar, Dec 07 2011
MATHEMATICA
a[n_, k_] := (-1)^(n-k)*((n+1)*(n+1)!/((k+1)*(k+1)!))*Binomial[n, k]*HypergeometricPFQ[{-(n-k), k, 1}, {-(n+1), k+2}, 1]; Table[a[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 21 2013 *)
CROSSREFS
KEYWORD
AUTHOR
Wolfdieter Lang, Dec 06 2011
STATUS
approved