MODELS OF q-ALGEBRA REPRESENTATIONS:
q-INTEGRAL TRANSFORMS AND \ADDITION THEOREMS"
E.G. KALNINSy AND WILLARD MILLER, Jr.z
Abstract. In his classic book on group representations and special functions
Vilenkin studied the matrix elements of irreducible representations of the Euclidean
and oscillator Lie algebras with respect to countable bases of eigenfunctions of the
Cartan subalgebras, and he computed the summation identities for Bessel functions
and Laguerre polynomials associated with the addition theorems for these matrix
elements. He also studied matrix elements of the pseudo-Euclidean and pseudooscillator algebras with respect to the continuum bases of generalized eigenfunctions
of the Cartan subalgebras of these Lie algebras and this resulted in realizations of the
addition theorems for the matrix elements as integral transform identities for Bessel
functions and for con uent hypergeometric functions. Here we work out q-analogs of
these results in which the usual exponential function mapping from the Lie algebra
to the Lie group is replaced by the q-exponential mappings Eq and eq . This study of
representations of the Euclidean quantum algebra and the q-oscillator algebra (not
a quantum algebra) leads to summation, integral transform and q-integral transform identities for q-analogs of the Bessel and con uent hypergeometric functions,
extending the results of Vilenkin for the q = 1 case.
PACS: 02.20.+b, 03.65.Fd
1. Introduction. This paper is part of a series on the study of function space
models of irreducible representations of q-algebras [1-4]. These algebras and models
are motivated by recurrence relations satis ed by q-hypergeometric functions [511] and our treatment is an alternative to the theory of quantum groups [12-23].
In our earlier papers we considered irreducible representations of q-analogs of the
three-dimensional Euclidean Lie algebra and the four-dimensional oscillator algebra
(not a quantum algebra). We replaced the usual exponential function mapping
from the Lie algebra to the Lie group by the q-exponential mappings Eq and eq .
In place of the usual matrix elements on the group (arising from an irreducible
representation) we found several di erent types of matrix elements expressible in
terms of q-hypergeometric series. These q-matrix elements do not satisfy group
homomorphism properties, so they do not lead to addition theorems in the usual
sense, but to various q-analogs of addition theorems. All of the matrix elements
1991 Mathematics Subject Classi cation. 33D55, 33D45, 17B37, 81R50.
Key words and phrases. basic hypergeometric functions, q-algebras, quantum groups, integral
transforms.
yDepartment of Mathematics and Statistics, University of Waikato, Hamilton, New Zealand
zSchool of Mathematics and Institute for Mathematics and its Applications, University of
Minnesota, Minneapolis, Minnesota 55455. Work supported in part by the National Science
Foundation
under
grant
DMS
91{100324
1
Typeset by AMS-TEX
2
E.G. KALNINS AND WILLARD MILLER, JR.
are determined with respect to countable bases of eigenfunctions of the \Cartan
subalgebra" of the q-algebra.
In his classic book [24] Vilenkin studied the matrix elements of irreducible representations of the Euclidean and oscillator Lie algebras with respect to these same
countable bases, and he computed the identities for Bessel functions and Laguerre
polynomials associated with the addition theorems for these matrix elements. However, he also studied matrix elements of the pseudo-Euclidean and pseudo-oscillator
algebras with respect to the continuum bases of generalized eigenfunctions of the
Cartan subalgebras of these Lie algebras. (See also [25, 26] in these regards.) These
studies resulted in realizations of the addition theorems for the matrix elements as
integral transform identities for Bessel functions and for con uent hypergeometric
functions. Here we work out q-analogs of these results.
In x2 we introduce a family of four-parameter q-matrix elements for the unitary irreducible representations of the Euclidean Lie algebra with respect to the
standard countable eigenbasis and work out an associated \addition theorem" for
these matrix elements". (These functions were introduced earlier in [27] as generating functions for q-Bessel functions but their role as matrix elements obeying an
\addition theorem" was not pursued. Simultaneously with the issuance of a rst
preprint of our results Koelink [28] issued a preprint in which he proved this same
addition theorem and reinterpreted it to yield a q-analogue of an integral of Weber
and Sonine and of the Fourier-Bessel transform.) Then, in analogy with Vilenkin's
work for true group representations, we introduce a q-analog of matrix elements of
the pseudo-Euclidean group with respect to a continuum basis of generalized eigenfunctions. This study involves use of the Mellin transform and leads to integral
transform identities for q-Bessel functions, interpreted as \addition theorems" that
in the limit as q ! 1 go to identities derived by Vilenkin. In x3 we introduce a different q-analog of the pseudo-Euclidean group and apply the same procedures. This
time it is the complex Fourier series that is relevant and the \addition theorems"
lead to q-integral transform identities for q-Bessel functions.
In x3 and x4 we apply the same ideas to q-analogs of the oscillator and pseudooscillator algebras (these are not quantum groups) and obtain discrete, integral
transform and q-integral transform identities for q-analogs of the con uent hypergeometric functions, extending the results of Vilenkin for the q = 1 case.
The notation used for q-series and q-integrals in this paper follows that of Gasper
and Rahman [29].
2. Matrix elements of m(2) representations. The three dimensional Lie algebra m(2) is determined by its generators H , E+ , E which obey the commutation
relations
(2.1)
[H; E+ ] = E+ ; [H; E ] = E ;
[E+ ; E ] = 0:
We consider irreducible representations (! of m(2), characterized by the positive number !. The spectrum of H corresponding to (!) is the set Z = fm :
man integerg and the complex representation space has basis vectors fm , m 2 Z ,
such that
(2.2)
E fm = !fm1 ; Hfm = mfm ; E+ E fm = !2 fm;
q
-INTEGRAL TRANSFORMS AND ADDITION THEOREMS
3
where C E+ E is an invariant operator. A simple realization of (!) is given by
the operators
d ; E = !z; E = !
(2.3)
H = m0 + z dz
+
z
acting on the space of all linear combinations of the functions z n , z a complex
variable, n 2 Z , with basis vectors fm (z ) = z m.
We can introduce an inner product such that < fn ; fn0 >= nn0 , n; n0 2 Z . On
the dense subspace K of all nite linear combinations of the basis vectors we have
(2.4)
< E+ f; f 0 > = < f; E f 0 >; < Hf; f 0 > = < f; Hf 0 >;
for all f; f 0 2 K, so H = H and E+ = E . In terms of the operators (2.3) we can
obtain a realization of (!) and its Hilbert space structure by setting z = ei :
d ; E = !ei ; E = !e i ;
H = i d
(2.5)
+
2
fn (z ) = ein ; < f; f 0 >= 21
f (ei )f 0 (ei ) d:
0
Matrix elements Tm0 m of the complex motion group in the representation (!)
are typically de ned by the expansions
Z
e E+ e E e H fm =
(2.6)
X1
m0 = 1
Tm0 m ( ; ; )fm0 ;
[2, 10, 30]. The group multiplication property of the operators on the left-hand
side of (2.6) leads to addition theorems for the matrix elements. For convenience
in the computations to follow we shall limit ourselves to the case where = 0.
With the q-analogs of the exponential function
1
k
eq (x) = (qx; q) = (x; 1q) ; jxj < 1;
k
1
k=0
1 qk(k 1)=2 xk
(2.7)
= ( x; q)1 ;
Eq (x) =
k=0 (q ; q )k
we employ the model (2.3) to de ne the following q-analogs of matrix elements of
(!), [2]:
X
X
X1 Tne;en ( ; )fn ; j! j; j! j < 1
n 1
1
X
b) eq ( E )Eq ( E )fn =
Tne;E
n ( ; )fn ; j! j < 1
n 1
X1 TnE;en ( ; )fn ; j! j < 1
c) Eq ( E )eq ( E )fn =
n 1
1
X
(2.8)
d) Eq ( E )Eq ( E )fn =
TnE;E
n ( ; )fn ;
n 1
X1 Tn n( ; ; ; )fn ;
e) eq ( E )Eq ( E )eq ( E )Eq (E )fn =
a) eq ( E+ )eq ( E )fn =
+
+
+
+
+
(
0
(
0
)
0
(
0
)
0
(
0
0=
0=
0=
0=
n0 = 1
)
0
)
0
j! j < jz j < 1=j! j:
0
0
4
E.G. KALNINS AND WILLARD MILLER, JR.
Here, 0 < q < 1 and ; ; ; 2 /C . (All of these matrix elements, except (2.8e),
were studied in [2].) Since E+ = E we have
T(
e;e)
n0 n
(2.9)
( ; ) = < e ( E+ )e ( E )f ; f >=< f ; e ( E+ )e ( E )f >
=T ( )( ; ) = T ( ) ( ; );
q
q
e;e
e;E )
n0 n
n
n0
q
q
e;e
nn0
T(
n0
n
nn0
( ; ) = T(
E;e)
nn0
(2.10)
T
n0 n
( ; ); T (
E;E )
n0 n
( ; ) = T(
E;E )
nn0
( ; );
( ; ; ; ) = T ( ; ; ; ):
nn0
Furthermore, since e (x)E ( x) = 1, we have the identities
q
1
X
(2.11)
`=
1
1
X
(2.12)
`=
1
T(
q
e;e)
n0 `
T(
( ; )T (
e;E )
n0 `
(
;
) = ; j! j; j! j < 1
(
;
) = ; j! j; j! j < 1;
E;E )
`n
( ; )T (
E;e)
`n
n0 n
n0 n
and, of primary interest here:
T
n0 n
( ; ; ; ) =
1
X
T(
( ; )T (
T(
(; )T (
e;e)
1
1
X
n0 `
E;E )
`n
(; ); j !j; j !j < 1;
`=
(2.13)
=
`=
1
X
`=
(2.14)
1
1
T ( ; ; ; )T ( 0 ;
n0 `
e;E )
n0 `
E;e)
`n
; 0;
`n
( ; ); j !j; j !j < 1;
) = T ( 0 ; ; 0 ; );
n0 n
0 0 !2 j; j 0 !2 j; j 0 !2 j < 1:
j ! j; j
2
(Note that our operator derivations of these formulas and of many formulas to
follow lead automatically to formal power series identities in the `group parameters'.
These identities must then be examined case by case to determine when the series
are convergent as analytic functions of the group parameters.) Using the model
(2.3) to treat (2.8) as generating functions for the matrix elements and computing
the coecients of z in the resulting expressions we obtain the explicit results:
(q +1 ; q)1 ( !)
0;
0 ; q; !2
(2.15)
T ( )( ; ) =
2 1
+1
q
(q; q)1
+1
= (q (q; q) ;(q)1!(2 ;!q)) 0 1 q +1 ; q; !2 q +1 ;
1
1
0
n
n
e;e
0
n
n
0
n
n0 n
n
n
0
n
n
0
n
0
n
n
0
n
n
n
0
q
(2.16)
(e;E )
Tn n
0
-INTEGRAL TRANSFORMS AND ADDITION THEOREMS
n
( ; ) = (q
; q)1 ( !)n
(q; q)1
n0 +1
n
= (q
n0
q
(n
; q)1 ( !)n
(q ; q )1
n+1
0
(2.17)
0
n0 )(n n0
1)=2
n
1
1
1 q n
0
1 qn
n+1
0
0
n
; q;
0
!
+1
2
5
; q;
!q
2
n n0
;
)
(e;E )
Tn(E;e
n ( ; ) = Tnn ( ; );
0
(2.18)
)
Tn(E;E
n ( ; )=
0
(qn
0
; q)1 ( !)n
(q; q)1
n0 +1
n0
q(n
n0 )(n n0
1)=2
0
1 q n
n +1 ; q;
0
!2 qn
n0
If 6= 0 we can express these results in terms of the Jackson q-Bessel functions
[28, page 25]
(q +1 ; q)1 ( z )
0; 0 ; q; z 2 ;
J(1) (z ; q) =
(q; q)1 2 2 1 q +1
4
(2.19)
J(2) (z ; q) =
(q +1 ; q)1 ( z )
z 2q +1
;
;
q;
0 1
q +1
(q; q)1 2
4
J(2) (z ; q) = ( z 2=4; q)1 J(1) (z ; q);
and the Hahn-Exton q-Bessel function [27]
(
q +1 ; q)1
0
2
J (z ; q) =
(q; q)1 z 1 1 q +1 ; q; qz :
(2.20)
Indeed, setting = irei , = ire
coordinates [r; ei ] we have
Tn n ( ; ) Tn n [r; e
(e;e)
0
(2.21)
(e;e)
0
i
)
i
Tn(e;E
n [r; e
)
i
Tn(E;e
n [r; e
)
i
Tn(E;E
n [r; e
0
0
0
i
, we see that in terms of the new complex
i( 2 + )(n n )
] = e( r2 !2 ; q) Jn(2)n (2r!; q)
1
] = ei( 2 )(n n) q(n n)=2 Jn n (r!q 21 ; q)
] = ei( 2 + )(n n ) q(n n )=2 Jn n (r!q 21 ; q)
2
] = ei( 2 + )(n n ) q(n n ) =2 Jn(2)n (2r!q 21 ; q):
0
0
0
0
0
0
0
0
0
0
0
(Note that J n (z ; q) = ( 1)nqn=2 Jn (zqn=2 ; q), J (2)n (z ; q) = ( 1)n Jn(2) (z ; q) for integer n.)
For the matrix elements (2.8e) we obtain (through the use of the q-binomial
theorem on the factors involving E+ and, separately, on the factors involving E )
(2.22)
Tn n ( ; ; ; ) =
0
( !)n
( = ; q)n
(q ; q )n n
n0
0
n0
2
1
qn
= ;
n
0
+1
qn
n0
= ; q;
!2
:
6
E.G. KALNINS AND WILLARD MILLER, JR.
n n
n n
= ( !) (q;(q) = ; q)n n 2 1 qnq n+1= ; = ; q; !2 ; j !2 j < 1:
n n
Alternatively, we could obtain this result by writing the matrix element as a contour
integral
0
0
0
0
0
I
1
( !z ; q)1 ( !=z ; q)1 z n
Tn n ( ; ; ; ) =
2i
( !z ; q)1 ( !=z ; q)1
(2.23)
0
n0
1 dz;
where the contour is the unit circle centered at the origin of the complex z-plane,
and evaluating the integral by residues. Setting = (1 q) 0 ; ; = (1 q)0 we
see that in the limit as q ! 1,
0
0
(2.24) Tn n ( ; ; ; ) ! [!((n + n 0)]+ 1) 0 F1 n n0 + 1 ; ( + )( + )!2 ;
n n0
0
expressible in terms of ordinary Bessel functions, [24].
The second of equations (2.13) was already derived by Koornwinder and Swarttouw [27, Proposition 4.1] where it was interpreted as a q-analog of Graf's addition
formula for Bessel functions:
y x=s
y xs
n=2
Jn
p
(y x=s)(y xs) =
1
X
k=
1
sk Jn+k (y)Jk (x):
The rst equation (2.13) and (2.14) have similar interpretations.
The \addition theorem" (2.14) reads, [28],
(2.25)
( 0 !)n
( = 0 ; q)n n
= 0;
qn n = 0 ; q; 0 0 !2 =
2
1
n
n
+1
(q; q)n n
q
1
`
n
` n =
X ( !) ( = ; q)` n
=
;
q
2
; q; !
2 1 q` n +1
(q; q)` n
`= 1
` n 0
( 0 !)` n( = 0 ; q)` n
q = ; = 0 ; q; 0 0 !2 ;
2 1 q` n+1
(q; q)` n
j !2 j; j 0 0 !2 j; j 0 !2j; j 0 !2 j < 1:
n0
0
0
0
0
0
0
0
0
0
Next we introduce a model of a q-analog of the pseudo-Euclidean group. The
model consists of a Hilbert space of complex valued functions f (x) f^() where
x = e and is a real variable, such that jjf jj2 < 1 and the inner product is
(2.26)
< f; g >=
Z1
0
f (x)g (x)
Z1
dx
f^()^g () d;
=
x
1
and jjf jj2 =< f; f >. The formal action of the q-algebra is
!
d
d
E+ = !x = !e E = = !e ; H = x = :
x
dx d
q
-INTEGRAL TRANSFORMS AND ADDITION THEOREMS
7
The action of the \pseudo-Euclidean group" is given by the formal operator T( ; ; ; )
where
(2.27)
T( ; ; ; )f^() = eq ( E+ )Eq ( E+ )eq ( E )Eq (E )f^()
!e ; !e ; q)1 f^():
= (( !e
; !e ; q)1
We require that neither or is negative, so that the denominator in (2.27)
never vanishes. Then for various values of the parameters ; ; ; the operator
T corresponds to multiplication by a bounded function of . In particular, the
multiplier is bounded under the following circumstances:
8 1) j = j < 1; j= j < 1
>
>
>
< 2) j = j < 1; = = 0
>
3) j= j < 1; = = 0
>
>
: 4) = = = = 0:
For these estimates we make use of the identities
(2.28)
nn
Aq
(Aq n ; q)1 = ( A)n q
( Aq ; q)n (A; q)1 ; ((Bq
( +1)
2
n ; q )1 A ( q ; q )n (A; q)1
A
n ; q )1 = B ( q ; q )n (B ; q )1 :
B
Following [24, Chapter 5], we will compute the matrix elements of the operator T
with respect to a continuum basis in which H is diagonalized. We rst restrict our
attention to the subspace G of the Hilbert space where G consists of those functions
f^ that are C 1 with compact support. Then as shown in [24, 26], the complex
Fourier or Mellin transform
Z1
Z1
f^()e d =
f (x)x 1 dx
(2.29)
F () =
1
0
has the properties that 1) F () is an entire (analytic) function of , 2) jF ()j <
CekjRe j for some positive constants C; k and 3) F decreases rapidly on every
straight line parallel to the imaginary axis in the complex -plane, i.e., limt!1 jtjn jF (c+
it)j = 0 for n = 0; 1; 2; .. (We denote the space of transforms of functions in G
by G^.) Furthermore we have the inversion formula
Z a+i1
1
^
(2.30)
f () = 2i
F ()e d;
a i1
for any real number a.
Now, the induced action of the operator q H on the transformed functions F
becomes diagonal:
Z1
Z1h
i
f ( + )e d = e F ():
q H F ()
q H f^() e d =
1
1
8
E.G. KALNINS AND WILLARD MILLER, JR.
Furthermore, the induced action of the operator T on G^ is given by
Z 1 ( !e ; !e ; q)
1 f^()e d;
T( ; ; ; )F () =
1 ( !e ; !e ; q)1
or
Z
Z 1 ( !e ; !e ; q)
1 d a+i1 F ()e(
T( ; ; ; )F () = 21i
a i1
1 ( !e ; !e ; q)1
)
d:
If j qa = j < 1; jq a = j < 1, then the iterated integral is absolutely convergent
and we can interchange the order of integration to obtain
Z a+i1
(2.31)
T( ; ; ; )F () =
K (; ; ; ; ; )F () d;
a i1
where
Z 1 ( !e ; !e ; q)
1
1 ( ) d
K (; ; ; ; ; ) =
2i 1 ( !e ; !e ; q)1 e
Z 1 ( !x; !=x; q)
1
1 x 1 dx:
= 2i
(
!x;
!=x
;
q
)
1
0
To compute the kernel function K we evaluate the contour integral
I
1
( !z; !=z ; q)1 1 dz
(2.32)
IN;M =
2i C ( !z; !=z ; q)1 z
N;M
along the closed contour CN;M on the Riemann surface of the integrand, where
N; M are positive integers and the contour is made up of the curves
CN;M
8
1)
>
>
>
>
< 2)
>
3)
>
>
>
: 4)
z = t;
!qM +1=2 t 1! q N 1=2
z = 1! q N 1=2 eit ; 0 t 2
1
N 1=2 t !q M +1=2
z = e2i t;
!q
z = !qM +1=2 eit ; 2 t 0:
In the limit as N ! 1; M ! 1 the integrals on the large and on the small circle
go to zero if j q + = j < 1; jq = j < 1. Then, evaluating (2.32) by residues
and (temporarily) assuming that j !2 j < 1 to make use of Heine's transformation
[29, page 9], we obtain
1 Z 1 ( !x; !=x; q)1 x 1 dx =
K (; ; ; ; ; ) =
2i 0 ( !x; !=x; q)1
(2.33)
"
( ; q +1 ; q)1 1 ;
1
q
2
; q; !
2i sin ( ) (q; q ; q)1 ( ! ) 2 1 q +1
#
q
( ; q +1 ; q)1
;
2
( !) 2 1 +1
; q; ! :
+
q
(q; q ; q)1
q
-INTEGRAL TRANSFORMS AND ADDITION THEOREMS
9
(The apparent singularities at !2 = q n , n = 0; 1; , are removable.) The
following special cases of (2.33) are of interest:
( ; q +1 ; q)1
1
+ j < 1; jq =
K (; ; ; ; ; ) =
2i sin ( ) (q; q ; q)1 ( !) ; jq
(2.34)
( ; q +1 ; q)1 1
1
+ = j < 1; jq
K (; ; ; ; ; ) =
2i sin ( ) (q; q ; q)1 ( ! ) ; j q
" 1
2
( !)
1
!
+1
K (; ; ; 0; ; 0) =
2i sin ( )( !2 ; q)1 (q; ; q)1 1 1 0 ; q; q
2
(
! )
!
+1
:
+ (q; q) 1 1 0 ; q; q
1
Setting = (1 q) 0 ; ; = (1 q)0 in (2.33) we see that in the limit as
q!1 ,
[!( 0 0 )]
1
0
0 )( 0
K (; ; ; ; ; ) !
2i sin ( )
( + 1) 0 F1 + 1 ; (
[
!( 0 0 )]
0
0
0
0
2
(2.35)
)( )! :
+ ( + 1) 0 F1 + 1 ; (
From the expression (2.27) we have the \addition formula"
T( ; ; ; )T( 0 ; ; 0 ; ) = T( 0 ; ; 0 ; )
which, for the kernel functions, takes the form
(2.36)
Z b+i1
K (; ; 0 ; ; 0 ; )F () d =
b i1
Z b+i1
Z a+i1
K (; ; ; ; ; ) d
K (; ; 0 ; ; 0 ; )F () d;
b i1
j < 1;
j < 1;
0 )!2
a i1
see [24, page 268]. Then, if the integrals in (2.36) are absolutely convergent we
have the functional relations
Z a+i1
(2.37) K (; ; 0 ; ; 0 ; ) =
K (; ; ; ; ; )K (; ; 0 ; ; 0 ; ) d;
a i1
0
j
j; jq = j < jqa j < j q = j; j 0 q = j:
Two special cases of (2.37) are of particular interest. If = ; 0 = we have
1 Z a+i1 ( !) ( 1! ) ( ; q +1 ; ; q +1 ; q)1 d;
K (; ; 0 ; ; ; ) =
4 a i1 sin ( ) sin ( ) (q; q ; q; q ; q)1
jq = j; j q = 0 j < qa < jq j; jq j;
and if just = 0 we have
Z a+i1
( 1! ) ( ; q +1 ; q)1
K (; ; 0 ; ; ; ) =
K (; ; ; ; ; )
2i sin ( ) (q; q ; q)1 d:
a i1
q =
0
0
0
0
0
0
10
E.G. KALNINS AND WILLARD MILLER, JR.
3. A discrete model of m(2) representations. In this section we study a model
of an alternate q-analog of the pseudo-Euclidean group. Here the generators of our
algebra are qH , E+ , E which obey the relations
(3.1)
qH E+ = qE+ qH ; qqH E = E qH ; [E+ ; E ] = 0:
We consider the following class of irreducible representations (!) for this algebra,
characterized by the positive number !. The Hilbert space consists of complex
functions f (x) with domain x = qn ; n = 0; 1; 2; and such that (f; f ) < 1,
where the inner product is
(3.2)
(f; g) =
1
X
n= 1
f (qn )g(qn ):
The action of the algebra on this Hilbert space is given by the operators
(3.3)
E+ = !x; E = !x ; qH f (x) = f (qx):
To de ne these operators rigorously we can restrict their action to, say, the dense
subspace L of all functions in the Hilbert space that are nonzero at only a nite
number of points. Then it is easy to show that E+ = E+ ; E = E ; (qH ) =
(qH ) 1 . We de ne the (inverse) Fourier transform F of f 2 L by
(3.4)
F() F [z ] = (f; x ) =
1
X
n= 1
f (qn )qn ; 2 C ;
where z = q . Then the induced action of the algebra on the transform space L^ is
qH F [z ] = z 1F [z ]; E F [z ] = !F [q1 z ];
so the operator qH is diagonalized in the transform space. Clearly, every F 2 L^ is
analytic for all z 6= 0. We can recover f from its transform F via the formula
(3.5)
I
f (qm ) = 21i F [z ]z m 1 dz
where the integration path is a simple closed curve around the origin in the z -plane.
Now the action of the \pseudo-Euclidean group" is given by the operator T( ; ; ; )
where
(3.6)
T( ; ; ; )f (x) = eq ( E+ )Eq ( E+ )eq ( E )Eq (E )f (x)
!x; !=x; q)1 f (x); f 2 L:
= (( !x;
!=x; q)
1
The induced action of T on L^ is given by
(3.7)
I
T( ; ; ; )F [z] = K (z=w; ; ; ; )F [w] dw
w;
q
-INTEGRAL TRANSFORMS AND ADDITION THEOREMS
where
11
1 ( !qn ; !q n; q) z n
1 X
1
2i n= 1 ( !qn ; !q n; q)1 w
Z1
( !x; !=x; q)1 x 1 d x; z = q ; w = q
= 2i(11 q)
q
(
!x; !=x; q)1
0
!; !; q)1
!; q=! ; q; z ;
= 21i (( !;
!; q= !
!; q) 2 2
w
K (z=w; ; ; ; ) =
(3.8)
1
and we require j j < j wz j < j j. Here
2 2
1
a1 ; a2 ; q; z = X (a1 ; a2 ; q)n z n;
b1 ; b2
n= 1 (b1 ; b2 ; q )n
Using Ramanujan's 1
(3.8):
1
j ab1ba2 j < jz j < 1:
1 2
sum, [29, page 239] we have the following special cases of
1 (q; = ; qz= !w; !w=z ; q)1
w
2i ( !; q= !; z= w; w=z ; q)1 ; j j < j z j < 1
1 (q; = ; !z=w; qw= !z ; q)1
z
K (z=w; ; ; ; ) =
2i ( !; q= !; z=w; w= z ; q)1 ; j j < j w j < 1:
Now the formula
K (z=w; ; ; ; ) =
T( ; ; ; )T( 0 ; ; 0 ; ) = T( 0 ; ; 0 ; )
leads to the functional relation
(3.9)
K (z=w; 0 ; ; 0 ; ) =
I
dy
K (z=y; ; ; ; )K (y=w; 0; ; 0 ; ) ;
y
where, choosing the integration path as the unit circle jyj = 1, we have the requirements j = j < jz j < j =j, j = 0j < jwj < j 0 = j.
Two special cases are of particular interest. If = ; 0 = we have
0 !; q=!
z
( !; !; q)1 2 2
!;
q= ! ; q; w
I
!y; !y=z; q; = 0; 0 !y=w; qw= 0 !y; q)1 dy
= 21i (q; =( ;q= qz=
!; z= y; y=z; !; q= 0!; y=w; w= 0 y; q)1
y
1 < jwj < j 0 = j; 1 < jz j < j =j;
and if just = 0 we have
0
!; q=! ; q; z
2 2
!;
q= !
w
I
!; q=! ; q; z (q; = 0 ; 0 !y=w; qw= 0 !y; q)1 dy ;
= 2 2
!; q= !
y ( !; q= 0!; y=w; w= 0 y; q)1 y
1 < jwj < j 0 = j; j = j < jz j < j =j:
12
E.G. KALNINS AND WILLARD MILLER, JR.
4. Matrix elements of oscillator algebra representations. In [1] a q-analog
of the oscillator algebra was introduced. This is the associative algebra generated
by the four elements H , E+ , E , E that obey the commutation relations
[H; E+ ] = E+ ; [H; E ] = E ;
(4.1)
[E+ ; E ] = q H E; [E; E ] = [E; H ] = 0:
It admits a class of algebraically irreducible representations "`; where `; are real
numbers and ` > 0. These are de ned on a Hilbert space H with orthogonal basis
ffn : n = 0; 1; g where
E+ fn = `q (n+1)=2 fn+1
1 qn f
E fn = `q n=2
1 q n 1
Hfn = ( + n)fn Efn = `2 q 1 fn :
(4.2)
Furthermore, the formal adjoints satisfy (E+ ) = E , H = H , E = E. The
elements C = qq H E + (q 1)E+ E and E lie in the center of this algebra, and
corresponding to the irreducible representation "`; we have C = `2 I , E = `2 q 1 I
where I is the identity operator on H.
A convenient model of "`;, [1], is determined by the orthonormal basis functions
(4.3)
en
= qn(n+1)=4
s (1
q )n n
(q ; q )n z ;
n = 0; 1; ;
(so that fn (z ) = qn(n+1)=4 z n) and the operators
= (1 ` q)z (1 Tz 1)
d
H = + z ; E = `2 q 1 I:
dz
E+ = `zI; E
(4.4)
The inner product is
(f; g) =
ZZ 1
1
where z = x + iy and
f (z )g (z )(z; z) dxdy
1 q
( (1 q)zz; q)1 ln q 1 :
The model Hilbert space H(z ) consists of all functions
(z; z) =
f 0 (z ) =
X1
n=0
cn z n
such that
X1 j
n=0
cn j2 q n(n+1)=2
(1 q)n < 1:
This is a space of entire functions; it has the kernel function
(4.5)
S (z 0 ; z ) =
X1 0 ( 0) 0 ( ) = ( (1
n=0
en z en z
q )qz 0 z ; q )1 :
q
-INTEGRAL TRANSFORMS AND ADDITION THEOREMS
13
Although the parameter is essential in the computation of tensor products of
pairs of irreducible representations "`; , [1,4], it disappears from the nal expressions for the matrix elements studied in this paper. Thus we henceforth set = 0.
Using the relations (2.3) we have the following q-analogs of the matrix elements of
"`;0:
XT
X
e ( E )E ( E )f = T
X
e ( E )E ( E )f = T
X
E ( E )e ( E )f = T
X
E ( E )e ( E )f = T
X
E ( E )E ( E )f = T
X
E ( E )E ( E )f = T
(e+; e ) : eq ( E+ )eq ( E )fn =
(e+; E ) :
q
(e ; E +) :
(4.6)
+
q
(E +; e ) :
+
q
(E +; E ) :
+
q
+
n
+
q
(E +;e
n0 n
n0
n
0
(E ; E +) :
q
q
+
n
0
)
( ; )fn ;
0
; )fn ;
)
0
( ; )fn ;
0
; )fn ;
(E ;e+)
(
n0 n
n0
n
( ; )fn ;
(e ;E +)
(
n0 n
n0
n
)
(e+;E
n0 n
n0
n
q
q
n0
n
q
q
(E ; e+) :
q
(e+;e
n0 n
(E +;E
n0 n
)
0
( ; )fn ;
0
; )fn :
(E ;E +)
(
n0 n
n
0
(e+; E ; e ; E +) : eq (E+ )Eq ( E )eq ( E )Eq ( E+ )fn =
XT
n0
0
n0 n (
; ; ; )fn :
0
(The series for the matrix elements Tn(en ;e+) ( ; ) does not converge.) All of these
sets of matrix elements were studied in [4], with the exception of (e+,E-,e-,E+)
which will be the focus of attention here.
Since E+ = E the following relationships hold:
0
(e+;e )
Tn(en+;e )( ; )An n = Tnn
( ; );
0
0
0
(4.7)
(E +;e )
(E
Tn(en+;E )( ; )An n = Tnn
( ; ); Tn(en ;E+)( ; )An n = Tnn
0
0
0
(E +;E )
( ; ); Tn(En
Tn(En+;E )( ; )An n = Tnn
0
0
0
0
Here
0
0
;E +)
(E
( ; )An n = Tnn
0
0
An n = ((qq;;qq))n (1 q)n n :
0
0
0
n
Since eq (z )Eq ( z ) = 1, we have the identities
XT
X
b) T
a)
h
h
(e+;e
n0 h
)
(E
( ; )Thn
(E ;e+)
(
n0 h
;E +)
;e+)
0
( ;
(E +;e )
; )Thn
( ;
) = n n
0
) = n n :
0
( ; );
;E +)
( ; ):
14
E.G. KALNINS AND WILLARD MILLER, JR.
Using the model (4.4) to compute the matrix elements (which are model independent) we obtain the explicit results
Tn(en+;e ) ( ; )
n
n n+1 ; q )1 ( `)n n
`2
; 0 ; q;
= (q
q(n n )(n+n +1)=4 2 1 qnq n+1
(q; q)1
1 q
n
n
n
+1
n
+1
n
n
`2
(
q
;
q)1 (q ; q)1 ( `)
q
;
0
(n n)(n +n+1)=4
= (q; q) (qn+1 ; q) (1 q)n n q
; q; 1 q
2 1
qn n +1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Tn(En+;e )( ; )
(
qn +1 ; q)1 (qn n +1 ; q)1 ( `)n n (n n)(n +n+1)=4
q
= (q; q) (qn+1 ; q) (1 q)n n q
1 1
n
q
1
1
n
n
+1
n
; q)1 ( `) n q(n n)(n 3n 3)=4
q n ; q;
= (q
1 1
n
q n+1
(q; q)1
0
0
0
0
0
0
0
0
0
0
0
0
Tn(En+;E )( ; )
n n+1 ; q )1 ( `)n n
= (q
q(n n)(n
(q; q)1
n0
n0 +1 ; q; 1
`2 q n n
1 q
0
!
`2
q
0
0
0
0
0
n n +1
n +1 ; q )1 ( `)n
= (q (q; q) ;(qq)n1+1(q; q) (1
q)n n
1
1
0
n0
0
0
3
n
q
=
3) 4
1
`2 q n n
1 q
q n
0
2 qn n+1 ; 0 ; q;
0
n n0 )(n 3n0
(
=
3) 4
1
2
q n
0
; q;
qn n +1 ; 0
0
!
`2 q n n
1 q
Tn(En ;E+)( ; )
2
( (1 q`)q ; q)1 (qn n+1 ; q)1 ( `)n n (n n)(n 3n 3)=4 q n ; 0
`2
=
;
q;
q
2 1
qn n+1
(q; q)1
(1 q)q
2
`
n
n
+1
n
+1
n
n
(
; q)1 (q
; q)1 (q ; q)1 ( `)
q(n n )(n 3n 3)=4
= (1 q)q (q; q) (qn+1 ; q) (1 q)n n
1
1
n
`2
q
;
0
; q; (1 q)q :
2 1
qn n +1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
The matrix elements T (e+;e ); T (E+;e ); T (E+;E ) are polynomials in and
and the matrix elements T (E ;E+) are entire analytic functions of these variables.
In [4] it is shown that all of the remaining matrix elements (4.6), except the last,
can be expressed in terms of these four. Indeed, we have the operator identities
`2 q H 1
(4.8)
eq
eq ( E+ )Eq ( E ) = Eq ( E )eq ( E+ );
1 q
(4.9)
Eq ( E+ )eq ( E )eq
`2 q H 1
= eq ( E )Eq ( E+ )
1 q
0
!
q
-INTEGRAL TRANSFORMS AND ADDITION THEOREMS
which imply
TnE0 n ;e+) (
(4.10)
; ) = eq
(
Tn(en ;E+) ( ; ) = eq
15
!
`2 q n 1 (e+;E )
Tn n
( ; );
1 q
`2 q n 1 (E+;e )
( ; ):
Tn n
1 q
0
0
0
0
Thus the matrix elements Tn(En ;e+) are well de ned for j `2 q n 1 =(1 q)j < 1
and the matrix elements Tn(en ;E+) are well de ned for j `2 q n 1 =(1 q)j < 1.
From identities (4.8), (4.9) we can express the (e+; E ; e ; E +) operator in the
alternate forms
0
0
0
T( ; ; ; ) = eq (E )Eq ( E )eq ( E )Eq ( E )
+
H
= eq (E+ )Eq ( E+ )Eq ( `1 q q
2
(4.11)
+
1
2
H 1
)Eq ( E )eq ( E )eq ( `1 q q ):
An explicit computation of the matrix elements yields
Tn n ( ; ; ; ) =
0
(
1
(1
`2 q n
q
`2 q n
q
(4.12)
3
=
(
1
(1
3
; q )1 ( ; q )n n n
(`)
; q)1 (q; q)n n
0
1
2
n0
qq
`2 q n
q
j `q
n
2
q n
;
0
`2 q n
q
1
1
2
; q; 1 ` q
; q)1 ( ; q)n n (q; q)n ` n
1 ; q ) (q ; q )
1 q
1
n n (q ; q )n
1
0
qn n ;
1
`2 q n 0
q
q n
0
1
=(1 q)j < 1:
!
q(n+n +1)(n n)=4
0
0
!
0
0
1
n0
0
;
0
n
n
q +1 ;
2
n q (n+n0 +1)(n n0 )=4
0
0
n n;
q
0
qn n+1 ;
`2
1
2
; q; 1 ` q ;
Indeed, from (4.11) and the fact that E+ = E we have the identity
Tn n ( ; ; ; ) = (1
0
2
(q; q)n ( 1`q q n0
0
n
n
q)
(q; q)0n ( 1 `q2 q n
; q )1
Tnn ( ; ; ; ):
; q )1
1
1
0
Using Sears' 3 2 transform [29, page 61] we have the alternate form
2
( 1 `q q 1 ; q)1 ( ; q)n n (q; q)n ` n n (n+n +1)(n
q
Tn n ( ; ; ; ) =
( 1 `q2 qn n 1 ; q)1 (q; q)n n (q; q)n 1 q
!
q n n +1 ;
qn n ; q n
; q; q :
3 2
1 q n n +2
n n +1
0
0
0
0
0
0
0
0
q
0
0
;
`2 q
0
0
0
n)=4
16
E.G. KALNINS AND WILLARD MILLER, JR.
Setting
q!1 ,
= (1 q) 0 ; ; = (1 q)0 in (4.12) we see that in the limit as
[`( 0 + 0 )]n n e`2 ( +
Tn n ( ; ; ; ) !
(n0 n)!
if n0 n;
n n n `2
0
0
! [`( + )]
e
n0
0
0
0
0
0
0
)
0
1
(
F1 n0
0
+
if n n0 :
0
)
1
F1
n;
2
0 0 0 0
n + 1 ; ` ( + )( + ) ;
In order to derive identities for the matrix elements of the operators T( ; ; ; ),
(4.11) we cannot just multiply two general operators of this sort, as was done in
the previous sections. Indeed, the resulting formal power series diverge. However,
just as in (2.13) we can write T as a composition of two two-factor operators and
obtain the identities
1
X
(e ;E +)
(4.13)
Tn n ( ; ; ; ) = Tn(ek+;E )( ; )Tkn
( ; );
0
0
k=0
Tn n ( ; ; ; ) =
0
1
X
k=0
(E ;E +)
Tn(ek+;e )( ; )Tkn
( ; )
0
j ` =qn (1 q)j < 1:
2
+1
(Once the common factor eq ( `2 q n 1 =(1 q)) is removed from both sides of
the rst equation (4.13), this formula holds for all values of the parameters.) Note
that relations (2.10) are special cases of (4.13). These identities are q-analogs of an
addition formula for the con uent hypergeometric functions [30, Chapter 4]:
0
(n + n0 )! e [ ( + )]n F
n ; ( + )( + ) =
1 1
n+1
n!
1
X
( )j n j !
n0 ;
n n0 ; :
F
F
1 1
1 1
0
0
j n0 + 1
j n n0 + 1
j =0 (j n + 1) (j n n + 1)
0
5. Continuum bases for oscillator representations. Next we introduce a
model of a q-analog of the pseudo-oscillator group. The model consists of a Hilbert
space of complex valued functions f (x) where x is a positive real variable, such that
jjf jj2 < 1 and the inner product is
Z1
dx
f (x)g (x)
(5.1)
< f; g >=
x
0
and jjf jj2 =< f; f >. The formal action of the q-algebra is
E+ = `xI; E =
(5.2)
H =x
`
1
(1 q)x (1 Tx )
d
; E = `2 q 1 I:
dx
n0 ; ; `2 ( 0 + 0 )( 0 + 0 ) ;
n n0 + 1
q
-INTEGRAL TRANSFORMS AND ADDITION THEOREMS
17
The action of the \pseudo-oscillator group" is given by the formal operator T( ; ; ; )
where
(5.3)
T( ; ; ; )f (x) = eq (E )Eq ( E )eq ( E )Eq ( E )f (x)
=
`x; (1
`x;
(1
+
` ;q
1
X
q)x
1
` ;q
q(1
q)x
1 n=0
`
2
n
+
q ;
`x
q)
q; (1
;q
n Tx n f (x):
` ;q
q)x
n
(To derive this expression we have made use of the q-Gauss formula [29, page 10].)
We require that neither or is positive, so that (5.3) is well de ned when acting
on the subspace G of the Hilbert space, where G consists of those functions f (x) that
are C 1 with compact support in (0; 1). Indeed, for each such f the summation in
(5.3) is nite. Using the identity (2.28) we can show that if f 2 G has support in a
proper subset of the interval (0; K ] then the function T( ; ; ; )f (x) vanishes for
x K and if j = j < 1 this function remains bounded as x ! 0+.
Following [24, Chapter 8], we will compute the matrix elements of the operator
T with respect to a continuum basis in which H is diagonalized. We recall that the
Mellin transform of f 2 G,
(5.4)
F () =
Z1
0
f (x)x
1
dx
has the properties that 1) F () is an entire (analytic) function of , 2) jF ()j <
CekjRe j for some positive constants C; k and 3) F decreases rapidly on every
straight line parallel to the imaginary axis in the complex -plane. (We denote the
space of transforms of functions in G by G^ .) Furthermore we have the inversion
formula
Z a+i1
1
f (x) =
2i a i1 F ()x d;
for any real number a.
Clearly, the induced action of the operator q H on the transformed functions F
is diagonal:
q H F () = e F ():
Furthermore, the induced action of the operator T on G^ is given by (assuming that
neither or is positive and that j = j < 1),
T( ; ; ; )F ()
=
Z1
0
q
1
`x; (1 `q)x ; q X
`2 n `x ; ; q n
1
f (q n x)x
`
`
q
(1
q
)
`x; (1 q)x ; q n=0
q; (1 q)x ; q
1
n
Re > 0;
1
dx
18
or
E.G. KALNINS AND WILLARD MILLER, JR.
Z1
T( ; ; ; )F () = 1
`x; (1 `q)x ; q
1 dx
(1
1
q
Z
1
n
a+i1
;q
X
`x ;
`2
n
F ()qn x
`
q
(1
q
)
a i1
q; (1 q)x ; q
n=0
n
Z1
`x; (1 `q)x ; q
1 dx
= 1
2i
2i
Z a+i1
a i1
`x;
0
2
1
(1
(1
q ;
`x
`
q)x
` ;q
q)x
`x;
0
` ;q
q)x
1
2
; q; 1` q q
1
!
F ()x
1
d
1
d;
if the contour is chosen so that j `2 qa 1 =(1 q)j < 1. Using Heine's and Jackson's
transformations [29, page 241] we can write this result as
Z1
(
1
T( ; ; ; )F () =
2i
2
2
1
0
q ;
`2 q 1 ;
q
`x; q)1
`x; (1 `q)x ; q
dx
1
`2 q 1
1 q
` q ; q; (1
(1 q )x
qa < jq = j; jq j
Z a+i1
1
a i1
`2 q
q
1
!
`
F ()x
q)x
1
1
;
(1
`2 q
q
1
` q ; q
q)x
;q 1
1
d:
If, in addition, j = j <
< 1, then the iterated integral is
absolutely convergent and we can interchange the order of integration to obtain
(5.5)
T( ; ; ; )F () =
where
Z a+i1
a i1
K (; ; ; ; ; )F () d;
`2 q 1 ;
` q ; q
q
(1 q )x
1
2
`
`
1
`x; (1 q)x ; 1 q q ; q
1 !
2
`
1
q
q ;
`
1 q
1 dx;
2 2
`2 q 1 ;
` q ; q; (1 q )x x
1 q
(1 q )x
j = j < jq j < jq = j; jq j < 1; jq j < jq(1 q)= `2 j; ;
1 Z1
K (; ; ; ; ; ) =
2i 0
`x;
1
To compute the kernel function K we evaluate the contour integral
(5.6)
` q ; q
`2 q 1 ; q ) I
`z;
(
(1 q )z
1
1
1 q
1
IN;M =
2i ( 1 `q2 q 1 ; q)1 C
`z; (1 `q)z ; q
1
N;M
2
2
1
q ;
`2 q 1 ;
q
`2 q 1
q
` q ; q; (1
(1 q )z
1
!
`
z
q)z
1
dz;
nonpositive:
q
-INTEGRAL TRANSFORMS AND ADDITION THEOREMS
19
along the closed contour CN;M of (2.32). In the limit as N ! 1; M ! 1 the
integrals on the large and on the small circle go to zero. Then, evaluating (5.6) by
residues and using the formula
d
1 (a; q) (b; q) (c; q) xk yn (c; ax; by; q)
X
;
x;
y
1
k
n
n+k
c
= (d; x; y; q) 3 2 ax; by ; q; c ;
1
k;n=0 (d; q )n+k (q ; q )k (q ; q )n
see [31], we obtain
K (; ; ; ; ; ) =
1
2i sin ( )( 1`q2 ; q)1
(5.7)
2
2
q;
q +1 ;
(
; q +1 ; 1`q ; q)1
4( `)
2
3
2
(q; q ; q)1
q +1 ; 1`q
` ( ; q ; q +1 ; `2 q ; q)1
1 q
1 q
(q; q ; q ; q)1
q +1 ;
3 2 +1
q
;
q +1 ;
`2
1 qq
q
2
; q; 1 `q q 1
2
; q; 1 `q q 1
!#
!
:
(The apparent singularities at the zeros of ( 1`q ; q)1 are removable.) The following special cases of (5.7) are of interest:
2
1
2i sin ( )( 1`q2 ; q)1
(5.8)
+1
(
q +1 ; q)1
`2
q
( `)
(q; q)1 1 1 q +1 ; q; 1 q q
#
` (q ; q +1 ; q)
1 q +1 ; q; `2 q ;
1 1 q +1
1 q
(q; q)1
1 q
K (; ; 0; ; 0; ) =
jq j < 1 ;
(5.9)
K (; ; 0; ; ; 0) =
`
1
2i sin ( ) 1 q
( ; q ; q +1 ; q)1
(q; q ; q ; q)1
j = j < jq j < jq j < 1;
(5.10)
K (; ; ; 0; 0; ) =
( ; q +1 ; q)
1
(
`) 1 ;
2i sin ( )
(q; q ; q)1
jq j < jq j < jq = j:
20
E.G. KALNINS AND WILLARD MILLER, JR.
Setting
q!1 ,
= (1 q) 0 ; ; = (1 q)0 in (5.7) we see that in the limit as
[ `( 0 + 0 )]
1
+
1
2
0
0
0
0
K (; ; ; ; ; ) !
2i sin ( )
( + 1) 1 F1 + 1 ; ` ( + )( + )
[`( 0 + 0 )] () F
+ 1 ; `2 ( 0 + 0 )( 0 +
(5.11)
;
1
1
+1
() ( + 1)
in agreement with [24, Chapter 8; 25].
From the expression (5.3) we see that the \addition formula"
(5.12)
T( ; ; ; )T( 0 ; 0 ; ;
) = T( 0 ; 0 ; ; )
holds rigorously when both sides are applied to f 2 G, provided is not positive and
; ; 0 are not negative. Furthermore, one can show that for j = j < 1, the function
h(x) = T( ; ; ; )f (x) has the properties that x2 h(x); x2 h0 (x); x2 h00 (x) ! 0 as
x ! 0+. Thus, the Mellin transform H () of h, (5.4), has the properties that 1)
H () is an analytic function of for jq j < 1, 2) jF ()j < CekjRe j for some positive
constants C; k withjq 1 j < 1, and 3) limt!1 jtj2 jF (c + it)j = 0 for jq 2 j < 1.
For the kernel functions, (5.12) takes the form
(5.13)
Z b+i1
K (; ; 0 ; 0 ; ; )F () d =
b i1
Z b+i1
Z a+i1
K (; ; ; ; ; ) d
K (; ; 0 ; 0 ;
b i1
a i1
;
)F () d:
Then, if the integrals in (5.13) are absolutely convergent we have the functional
relations
Z a+i1
K (; ; ; ; ; )K (; ; 0 ; 0 ; ; ) d;
(5.14) K (; ; 0 ; 0 ; ; ) =
a i1
j q = 0 j; jq = j < jqa j < j q = j; j 0 q = j:
An interesting special case is
K (; ; ; ; ; ) =
or
K (; ; ; ; ; ) =
1
( `12 qq 1 ; q)1
Z a+i1
a i1
K (; ; ; 0; 0; )K (; ; 0; ; ; 0) d;
( ; ; q ; q)1
4(q; q; `12 qq 1 ; q)1
Z a+i1 ( `) ( ` ) (q +1 ; q +1 ; q ; `2 q 1 ; q)1
1 q
1 q
d;
;
;
sin
(
)
sin
(
)(
q
q
q ; q)1
a i1
j = j < jq j < jqa j < 1;
jq j < jqa j < jq = j:
q
-INTEGRAL TRANSFORMS AND ADDITION THEOREMS
21
We conclude by studying a model of an alternate q-analog of the pseudo-oscillator
algebra. Here the generators of our algebra are q H , E+ , E , E which obey the
relations
(5.15)
E+ q H = qq H E+ ; qE q H = q H E ; [E+ ; E ] = q H E;
[E; E ] = [E; q H ] = 0:
We consider the following class of irreducible representations "`;0 for this algebra,
characterized by the positive number `. The Hilbert space consists of complex
functions f (x) with domain x = qn ; n = 0; 1; 2; and such that (f; f ) < 1,
where the inner product is
(5.16)
(f; g) =
1
X
n= 1
f (qn )g(qn ):
The action of the algebra on this Hilbert space is given by the operators
(5.17)
E+ = `x; E =
`
1
H
1
(1 q)x (1 Tx ); q f (x) = f (q x):
To de ne these operators rigorously we can restrict their action to, say, the dense
subspace L of all functions in the Hilbert space that are nonzero at only a nite
number of points. We de ne the (inverse) Fourier transform F of f 2 L by
(5.18)
F() F [z ] = (f; x ) =
1
X
n= 1
f (qn )qn ; 2 C ;
where z = q . Then the induced action of the algebra on the transform space L^ is
q H F [z ] = z F [z ]; E+ F [z ] = `F [qz ];
`
z
1
E F [z ] =
1 q (1 q )F [q z ];
so the operator q H is diagonalized in the transform space. Clearly, every F 2 L^
is analytic for all z 6= 0. We can recover f from its transform F via the formula
(5.19)
1
f (q m ) =
2i
I
F [z ]z m 1 dz
where the integration path is a simple closed curve around the origin in the z -plane.
The action of the \pseudo-oscillator group" is given by the operator T( ; ; ; ),
(5.3) for f 2 L. The induced action of T on L^ is given by
(5.20)
I
;
T( ; ; ; )F [z ] = K (z; w; ; ; ; )F [w] dw
w
22
E.G. KALNINS AND WILLARD MILLER, JR.
where
`x; q(1`2qw) ; ; q 1
`x; q(1`2qw) ; q 1
!
w;
`
2 1
; q; (1 q)x x 1 dq x;
`2 w
q(1 q)
j = j < jwj < jz= j; jz j < 1; jwj < jq(1 q)= `2 j; ; nonpositive; w = q ; z = q :
Z1
K (z; w; ; ; ; ) = 2i(11 q)
0
To evaluate the q-integral expression for the kernel function K we use the identity
2
1
C
A; B ; q; z = ( ABz
C ; q)1 3 2 A; B ; 0 ; q; q
Cq
C
( Bz
Bz ; C
C ; q )1
ABz ; 0
( BC ; A; Bz ; q)1
z;
C
; q; q ;
+
C ; q )1 3 2 Bzq ;
Bz
(C; z; Bz
C
[29, page 245, III.34] and obtain
K (z; w; ; ;
(5.21)
2
`2w ; q ; q; q ; w ; q )1
(
(
`;
`z
;
q
)
q
(1 q )
` `z
1
; ) = 2i(`; q) nfty 4 `2 w q
q ; z; w ; q)1
i
( q(1 q) ; ` ; `w; `w
z
!
2w
`
w
w;
q(1 q) ;
z ; q; q
3 2
2
`w
w
q(1 q) ;
!#
`;
q; q
(w; 1 `q ; q)1
1 q
`
`z
+
q2 ;
`
q ; q; q :
(z; q`w ; 1 `q ; q)1 3 3
`w 1 q ; `
Here
3
3
1 (a ; a ; a ; q)
a1 ; a2 ; a3 ; q; z = X
1
2
3
n z n ; j b1 b2 b3 j < jz j < 1:
b1 ; b2 ; b3
a1 a2 a3
n= 1 (b1 ; b2; b3 ; q )n
Now the formula
T( ; ; ; )T( 0 ; 0 ; ;
leads to the functional relation
(5.22)
K (z; w; 0 ; ; 0 ; ) =
I
) = T( 0 ; 0 ; ; )
K (z; y; ; ; ; )K (y; w; 0; ; 0 ; ) dyy ;
where, choosing the integration path as the unit circle jyj = 1, we have the requirements
j w= 0 j; jz= j < 1 < j z= j; j 0w= j:
q-INTEGRAL TRANSFORMS AND ADDITION THEOREMS
23
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