VARIABLES. Contents 1. Preliminaries 1 2. One variable Special cases 8 3. Two variables Special cases 14 References 16

Transcription

1 q-generating FUNCTIONS FOR ONE AND TWO VARIABLES. THOMAS ERNST Contents 1. Preliinaries 1. One variable 6.1. Special cases 8 3. Two variables Special cases 14 References 16 Abstract. We use a ultidiensional extension of Bailey s transfor to derive two very general q-generating functions, which are q-analogues of a paper by Exton [7]. These expressions are then specialised to give ore practical forulae, which are q-analogues of generating relations for Karlssons generalised Kape de Fériet function. A nuber of exaples are given including q-laguerre polynoials of two variables. 1. Preliinaries The purpose of this paper is to continue the study of q-special functions by the ethod outlined in [3] and [4]. The paper is a q-analogue of Exton [7]. We begin with a few definitions. Definition 1. The power function is defined by q a e alog(q). We always use the principal branch of the logarith. The q-analogues of a coplex nuber a and of the factorial function are defined by: (1) {a} q 1 qa 1 q, q C\{1}, Date: June 10, Matheatics Subject Classification: Priary 33D70; Secondary 33C65 1

2 THOMAS ERNST () {n} q! n {k} q, {0} q! 1, q C, k1 Definition. The q-hypergeoetric series was developed by Heine 1846 as a generalization of the hypergeoetric series: a; q n b; q n (3) φ 1 (a, b; c q, z) z n, 1; q n c; q n with the notation for the q-shifted factorial (copare [9, p.38]) 1, n 0; n 1 (4) a; q n (1 q a+ ) n 1,,..., 0 which is introduced in this paper. n0 Reark 1. The relation to Watson s notation, which is also included in the ethod, is (5) a; q n (q a ; q) n, where 1, n 0; n 1 (6) (a; q) n (1 aq ), n 1,,..., 0 Definition 3. Furtherore, (7) (a; q) (1 aq ), 0 < q < 1. 0 (8) (a; q) α (a; q) (aq α ; q), a q α, 0, 1,.... Definition 4. In the following, will denote the space of coplex nubers od πi. This is isoorphic to the cylinder R log q eπiθ, θ R. The operator is defined by : C Z C Z (9) a a + πi log q. Furtherore we define (10) a; q n ã; q n.

3 q-generating FUNCTIONS FOR ONE AND TWO VARIABLES. 3 By (9) it follows that n 1 (11) a; q n (1 + q a+ ), 0 where this tie the tilde denotes an involutioninvolution which changes a inus sign to a plus sign in all the n factors of a; q n. The following siple rules follow fro (9). (1) ã ± b ã ± b, (13) ã ± b a ± b, (14) q a q a, where the second equation is a consequence of the fact that we work od πi. log q Definition 5. Generalizing Heine s series, we shall define a q-hypergeoetric series by (copare [8, p.4]): [ ] pφ r (â 1,..., â p ; ˆb 1,..., ˆb â1,..., â p r q, z) p φ r q, z (15) n0 ˆb 1,..., ˆb r â 1,..., â p ; q [ n 1, ˆb 1,..., ˆb ( 1) n q ) ] 1+r p (n z n, r ; q n where q 0 when p > r + 1, and { a, if no tilde is involved (16) â ã otherwise We will skip the â for the rest of the paper. Definition 6. The following generalization of (15) will soeties be used: (17) p+p φ r+r (a 1,..., a p ; b 1,..., b r q; z; (s 1 ; q),..., (s p ; q); (t 1 ; q),..., (t r ; q)) [ ] p+p φ a1,..., a p r+r q; z; (s b 1,..., b 1 ; q),..., (s p ; q); (t 1 ; q),..., (t r ; q) r a 1 ; q n... a p ; q [ n ( 1) n q ) ] 1+r+r p p (n 1; q n b 1 ; q n... b r ; q n z n p n0 r (s k ; q) n (t k ; q) 1 k1 k1 n,

4 4 THOMAS ERNST where q 0 when p + p > r + r + 1. Reark. Equation (17) is used in certain special cases when we need factors (t; q) n in the q-series. Definition 7. Let the q-pochhaer sybol {a} n,q be defined by n 1 (18) {a} n,q {a + } q. 0 An equivalent sybol is defined in [6, p.18] and is used throughout that book. See also [1, p.138]. This quantity can be very useful in soe cases where we are looking for q-analogues and it is included in the new notation. The following ultidiensional generalization of Bailey s transfor was given by Exton [5, p.139]. Theore 1.1. If (19) γ 1,..., n (0) β 1,..., n p 1 1,...,p n n δ p1,...,p n u p1 1,...,p n n v p1 + 1,...,p n+ n, 1,..., n p 1,...,p n0 then forally (1) α 1,..., n γ 1,..., n α p1,...,p n u 1 p 1,..., n p n v p1 + 1,...,p n+ n, β 1,..., n δ 1,..., n. We assue that α, δ, u, v are functions of 1,..., n only. The notation denotes a ultiple suation with the indices 1,..., n running over all non-negative integer values. Definition 8. We will use the following abbreviation A () (a); q n a 1,..., a A ; q n a j ; q n. The following notation will be convenient. (3) QE(x) q x. j1 When there are several q:s, we generalize this to (4) QE(x, q i ) q x i.

5 q-generating FUNCTIONS FOR ONE AND TWO VARIABLES. 5 If {x j } n j1 and {y j} n j1 are two arbitrary sequences of coplex nubers, then their scalar product is defined by (5) xy n x j y j j1 We will only need one q-lauricella function, which is defined by (6) Φ (n) D (a, b 1,..., b n ; c q; x 1,..., x n ) a; q n b 1 ; q 1... b n ; q n n j1 x j j c; q n n j1 1; q. j The following reduction theore is a q-analogue of Appell and Kapé de Fériet [, p. 116]. Theore 1.. (7) Φ (n) D (a, b 1,..., b n ; c q; x, xq b, xq b b 3,..., xq b... b n ) φ 1 (a, b b n ; c q, xq b... b n ). Proof. In the LHS of (7) we change suation indices to {k l } n l1, where (8) k l n s. sl By atrix inversion, this is equivalent to (9) l k l k l+1, 1 l n 1, n k n.

6 6 THOMAS ERNST (30) LHS a; q n b 1 ; q 1... b n ; q n x n n j q j(b +...+b j ) c; q n n i j1 1;q j a; q n 1 k1 j1 b j; q kj k j+1 b n ; q kn x k 1 n 1 j q(k j+1 k j )(b +...+b j ) q ( kn)(b +...+b n) c; q n 1 k1 j1 1;q k j k j+1 1;q kn k i a; q n 1 k1 j1 b j; q kj k j ; q kj+1 q (k j+1)( b j +1) b n ; q kn x k 1 k i c; q n 1 k1 j1 1 b j k j ; q kj+1 1;q kj 1;q kn n 1 q (k j+1 k j )(b +...+b j ) q ( kn)(b +...+b n) j a, b 1 ; q k1xk1 c, 1; q k1 k 1,...,k n 1 n j b j, k j 1 ; q kj q (k j)(1 b j b j 1 ) 1, 1 b j 1 k j 1 ; q kj b n 1, k n, b n + b n 1 ; q kn 1 q (k n 1)(1 b n b n 1 b n) 1, 1 b n k n, b n 1 ; q kn 1 a, b 1 ; q k1xk1 c, 1; q k1 k 1,...,k n n j b j, k j 1 ; q kj q (k j)(1 b j b j 1 ) 1, 1 b j 1 k j 1 ; q kj 1 b n b n 1 b n k n ; q kn 1 b n k n ; q kn a, b 1 ; q k1xk1 c, 1; q k1 k 1,...,k n n j b n + b n 1 + b n ; q kn q k n (b n+b n 1 ) b n ; q kn. b j, k j 1 ; q kj q (k j)(1 b j b j 1 ) 1, 1 b j 1 k j 1 ; q kj We begin with the case n 1.. One variable Theore.1. If C( 1 ) is any arbitrary function, then, forally (31) C( 1 ) d; q 1 t 1 (tq d+ 1 ; q) (t; q) d; q 1 t 1 1; q 1 C(p 1 ) 1 ; q p1 ( 1) p 1 q p (p 1 ). p 1 0

7 q-generating FUNCTIONS FOR ONE AND TWO VARIABLES. 7 Proof. In (1) put (3) α 1 C( 1 ), (33) u 1 1 1; q 1, (34) v 1 1 and (35) δ 1 d; q 1 t 1. Now (19) and (0) iply that (36) and (37) γ 1 β 1 1 p 1 0 p 1 1 d; q 1 t 1 1 p 1 0 C(p 1 ) 1; q 1 p 1 C(p 1 ) 1 ; q p1 1; q 1 ( 1) p 1 q p (p1 ). d; q p1tp1 1; q p1 1 p 1 0 p 1 0 d + 1 ; q p1 t p 1 1; q p1 d; q 1 t 1 1 φ 0 (d + 1 ; q, t) d; q 1 t 1 (tqd+ 1 ; q) (t; q). d; q p1 + 1 t p ; q p1 The proof is copleted by substituting (36) and (37) into (1). Theore.. If C( 1 ) is any arbitrary function of 1, then, forally (38) E q (t) C( 1 )t 1 (1 q) 1 t 1 (1 q) 1 1; q 1 Proof. Let d in (31). C(p 1 ) 1 ; q p1 ( 1) p 1 q p (p 1 ). p 1 0 The theores.1 and. are uch too general for any practical purposes when deriving generating functions for various classes of q- hypergeoetric polynoials. A ore convenient for is obtained by considering the following special case.

8 8 THOMAS ERNST (39) C( 1 ) (a), (f 1); q 1 ( x 1 ) 1 q θ( 1) (h), (g 1 ), 1; q 1, where θ( 1 ) is an arbitrary function. Theore.1 can be written as (40) The confluent for (a), (f 1 ), d; q 1 ( x 1 ) 1 (h), (g 1 ), 1; q 1 d; q 1 t 1 1; q 1 p ; q p1 q p (p 1 ). t 1 q θ(1) (t; q) d+1 (a), (f 1 ); q p1 (x 1 ) p 1 q θ(p 1) (h), (g 1 ), 1; q p1 (41) E q (t) (a), (f 1 ); q 1 ( x 1 ) 1 (h), (g 1 ), 1; q 1 t 1 (1 q) 1 q θ( 1) t 1 (1 q) 1 1; q 1 1 ; q p1 q p (p 1 ) p 1 0 (a), (f 1 ); q p1 (x 1 ) p 1 q θ(p 1) (h), (g 1 ), 1; q p1 follows siilarly fro theore...1. Special cases. In the special case θ( 1 ) 0 and A + F 1 + H + G, the LHS of (40) can be written as (4) (43) 1 (t; q) d A+F +1φ H+G+1 ((a), (f 1 ), d; (h), (g 1 ) q; tx 1 ; ; (tq d ; q)). Put A F G 0, H 1, θ( 1 ) 1. Fro (40) it follows that ( x 1 ) 1 d; q 1 t 1 q 1 h 1 ; q 1 1; q 1 (t; q) d d; q 1 t 1 1; q 1 p ; q p1 q p+p 1 (p 1 ). (x 1 ) p 1 h 1 ; q p1 1; q p1

9 q-generating FUNCTIONS FOR ONE AND TWO VARIABLES. 9 By a change of variables x 1 x 1 q h 1 1 (1 q) this is equivalent to (44) n0 {c} n,q L (α) n,q(x)t n {1 + α} n,q n0 {c} n,q q n +αn ( xt) n {n} q!{1 + α} n,q (t; q) c+n 1 (t; q) c 1φ (c; 1 + α q; xtq 1+α (1 q); ; (tq c ; q)). n This is a wellknown generating function for the Laguerre polynoials. Put A H 1, F G 0, θ( 1 ) ( 1 ) in (41). Then we obtain the following generating function for the little q-jacobi polynoials: (45) t n (1 q) n 1; q n n p0 a + b + n + 1, n; q p x p q np a + 1, 1; q p t n (1 q) n 1; q n P n (xq n 1 ; a, b q) E q (t) 1 φ 1 (a + b + n + 1; a + 1 q, xt(1 q)). (46) Denote [ 4φ 7 (α) 4 φ 7 (47) [ 4φ 7 (β) 4 φ 7 a+b+n+1,, a+b+n+, 1+a, 1+a, +a, +a, 1, 1, 1 a+b+n+1 a+b+n+ a+b+n+ a+b+n+,, a+b+n+3, +a, +a, 3+a, 3+a, 3, 3, 1 a+b+n+3 ] q, q(1 q) x t, ] q, q 3 (1 q) x t. Making use of the decoposition of a series into even and odd parts fro [13, p.00,08], we can rewrite (45) in the for (48) P n (xq n 1 ; a, b q)t n P n+1 (xq n ; a, b q)t n+1 + {n} n0 q! {n + 1} n0 q! [ E q (t) 4φ 7 (α) xt {1 + a + b + n} ] q 4φ 7 (β), {1 + a} q

10 10 THOMAS ERNST and replacing t in (48) by it, we obtain (49) ( 1) n t n P n (xq n 1 ; a, b q) ( 1) n t n+1 P n+1 (xq n ; a, b q) + i {n} n0 q! {n + 1} n0 q! [ (Cos q (t) + isin q (t)) 4φ 7 (α) ixt {1 + a + b + n} ] q 4φ 7 (β). {1 + a} q (50) (51) Next equate real and iaginary parts fro both sides to arrive at the generating functions ( 1) n t n P n (xq n 1 ; a, b q) {n} q! n0 Cos q (t) 4 φ 7 (α) + xtsin q (t) {1 + a + b + n} q {1 + a} q 4φ 7 (β) and n0 ( 1) n t n+1 P n+1 (xq n ; a, b q) {n + 1} q! Sin q (t) 4 φ 7 (α) xtcos q (t) {1 + a + b + n} q {1 + a} q 4φ 7 (β). 3. Two variables We can generalize (31) to two variables. Theore 3.1. If C( 1, ) is any arbitrary function of 1,, then, forally C( 1, ) d; q 1 + j1 k j; q j t 1+ k 1 + k ; q 1 + (5) (tq d+ 1 k ; q) (tq k ; q) p 1,p 0 d; q 1 + j1 k j; q j t 1+ q k 1 + k ; q 1 + j1 1; q j C(p 1, p ) j ; q ( 1) p 1+p QE( j1 k ( p ) p 1 + p ). j1 ( )

11 q-generating FUNCTIONS FOR ONE AND TWO VARIABLES. 11 Proof. In (1) put (53) α 1, C( 1, ), (54) u 1, QE( 3 4 k ) j1 1; q j, (55) v 1, q 1 4 and (56) δ 1, q d; q 1 + j1 k j; q j t 1+ k 1 + k ; q 1 +. Now (19) and (0) iply that (57) β 1, p 1,p 0 1, p 1,p 0 C(p 1, p ) j1 1; q j p j QE( 3( 4 p ) k ( p ) + 1( 4 + p ) ) 1, C(p 1, p ) j1 j; q ( ) j1 1; q ( 1) p 1+p QE( j k ( p ) + + 1p 1 + p ), j1 and

12 1 THOMAS ERNST (58) γ 1, p 1 1,p d; q p1+p j1 k j; q t p 1+p j1 1; q p j j k 1 + k ; q p1 +p QE( p + 3(p 4 ) k (p ) + 1( 4 + p ) ) d; q p p + j1 k j; q + j t p 1+p j1 1; q p j k 1 + k ; q p1 +p p 1,p 0 QE( p (k + )) d; q 1 + j1 k j; q j t 1+ k 1 + k ; q 1 + p 1,p 0 d ; q p1 +p j1 k j + j ; q t p 1+p j1 1; q p j k 1 + k ; q p1 +p QE( p (k + )) d; q 1 + j1 k j; q j t 1+ k 1 + k ; q 1 + Φ () D (d + 1 +,, k 1 + 1, k + ; k k + q; t, tq k ) d; q 1 + j1 k j; q j t 1+ k 1 + k ; q 1 + 1φ 0 (d ; q, tq k ) d; q 1 + j1 k j; q j t 1+ k 1 + k ; q (tq k ; q)d+1 +. The proof is copleted by substituting (57) and (58) into (1). Theore 3.. If C( 1, ) is any arbitrary function of 1,, then, forally (59) p 1,p 0 E q (tq k )C( 1, ) j1 k j; q j t 1+ (1 q) 1+ k 1 + k ; q 1 + j1 k j; q j t 1+ (1 q) 1+ q k 1 + k ; q 1 + j1 1; q j C(p 1, p ) j ; q ( 1) p 1+p QE( j1 k ( p ) p 1 + p ). j1 ( )

13 q-generating FUNCTIONS FOR ONE AND TWO VARIABLES. 13 Proof. Let d in (5). The theores 3.1 and 3. are uch too general for any practical purposes when deriving generating functions for various classes of hypergeoetric polynoials. A ore convenient for is obtained by considering the following special case. (60) C( 1, ) (a); q 1 + q θ( 1, ) j1 (f j); q j ( x j ) j (h); q 1 + j1 g j, 1; q j, where θ( 1, ) is an arbitrary function. Theore 3.1 can be written as (a); q 1 + q θ( 1, ) j1 (f j); q j ( x j ) j (h), k 1 + k ; q 1 + j1 g j, 1; q j (61) d; q 1 + j1 k j; q j t 1+ (tq d+ 1 k ; q) (tq k ; q) d; q 1 + j1 k j; q j t 1+ q k 1 + k ; q 1 + j1 1; q j p 1,p 0 (a); q p1 +p q θ(p 1,p ) j1 (f j), j ; q x p j j (h); q p1 +p j1 g j, 1; q ( ) QE( k ( p ) p 1 + p ). j1

14 14 THOMAS ERNST The following confluent for follows siilarly fro theore (3.). E q (tq k ) (a); q 1 + q θ( 1, ) (h), k 1 + k ; q 1 + j1 g j, 1; q j t 1+ (1 q) 1+ (f j ), k j ; q j ( x j ) j j1 (6) j1 k j; q j t 1+ (1 q) 1+ q k 1 + k ; q 1 + j1 1; q j p 1,p 0 (a); q p1 +p q θ(p 1,p ) j1 (f j), j ; q x p j j (h); q p1 +p j1 g j, 1; q ( ) QE( k ( p ) + + 1p 1 + p ). j Special cases. Put A F G 0, H 1, θ( 1, ) 1 in (61). Then (63) q 1 d; q 1 + t 1+ j1 ( x j) j k j ; q j h, k 1 + k ; q 1 + (tq k ; q)d+1 + j1 1; q j d; q 1 + j1 k j; q j t 1+ q k 1 + k ; q 1 + j1 1; q j p 1,p 0 j1 j; q (x j ) p j ( ) j1 1; q QE( + p j + 1 p 1 ). p j j1 1, QE( k ( p )) h;q p1 +p By a change of variables x 1 x 1 q h 1 1 (1 q), x x (1 q) this is equivalent to (64) q 1 +1(h 1) d; q 1 + t 1+ j1 ( x j) j k j ; q j (1 q) 1+ h, k 1 + k ; q 1 + (tq k ; q)d+1 + j1 1; q j d; q 1 + j1 k j; q j t 1+ k 1 + k, h; q 1 + L h 1 1,,k,q (x 1, x ),

15 q-generating FUNCTIONS FOR ONE AND TWO VARIABLES. 15 where L α 1,,k,q (x 1, x ) is the q-laguerre poynoial in two variables given by (65) L α 1,,k,q (x 1, x ) α + 1; q 1 + j1 1; q j QE( 1 p p k ( p )) 1, p 1,p 0 q p 1 +αp 1 j1 j; q (x j ) p j 1 + α; q p1 +p j1 1; q p j ( ) )(1 q) p 1+p. QE( j1 By letting d h, d k 1 + k and d in (64), we obtain q-analogues of eq. A19-A1 in [7]. Put F G H 0, A 1, θ( 1, ) 1 in (61). Then (66) q 1 a, d; q 1 + t 1+ j1 ( x j) j k j ; q j k 1 + k ; q 1 + (tq k ; q)d+1 + j1 1; q j d; q 1 + j1 k j; q j t 1+ q k 1 + k ; q 1 + j1 1; q j 1, p 1,p 0 a; q p1 +p j1 j; q (x j ) p j j1 1; q p j QE( 1 p 1 + QE( k ( p )) j1 ) + p j ) ( d; q 1 + t 1+ q a; q 1 + j1 k j; q j ( x j ) j 1, k 1 + k ; q 1 + j1 1; q j j1 j; q ( x j ) p j p 1,p 0 a ; q p1 +p j1 1; q QE( k (p ) 1 p p j + p 1 p (p 1 + p )( 1 + ) + p j ap j ). j1 This is a q-analogue of eq. A in [7]. The sybol denotes that the equality is purely foral.

16 16 THOMAS ERNST Put A F H 0, G 1, θ( 1, ) 1 (67) q 1 d; q 1 + t 1+ j1 ( x j) j k j ; q j k 1 + k ; q 1 + (tq k ; q)d+1 + j1 g j, 1; q j d; q 1 + j1 k j; q j t 1+ q k 1 + k ; q 1 + j1 1; q j j1 j; q (x j ) p j j1 g QE( 1 p 1 + j, 1; q j1 1, p 1,p 0 ( ) in (61). Then QE( k ( p )) + p j ). By a change of variables x j x j q g j 1 (1 q), j 1, this is equivalent to (68) q 1 d; q 1 + t 1+ j1 ( x j) j q j(g j 1) k j ; q j (1 q) 1+ k 1 + k ; q 1 + (tq k ; q)d+1 + j1 g j, 1; q j d; q 1 + j1 k j; q j t 1+ q k 1 + k ; q 1 + j1 1; q j 1, p 1,p 0 (1 q) p 1+p j1 j; q (x j ) p j q p j(g j 1) j1 g j, 1; q QE( 1 p 1 + QE( k ( p )) j1 ) + p j ). ( By letting k i g i, d g 1 + g, d and k i g i, d in (68 ), we obtain q-analogues of eq. A39-A4 in [7]. Acknowledgents. I want to thank Per Karlsson who gave soe valueable coents on the new ethod for q-hypergeoetric series, and who told e about Exton s paper. References [1] Álvarez-Nodarse, R., Quintero, N.R., Ronveaux A., On the linearization proble involving Pochhaer sybols and their q-analogues, J. Coput. Appl. Math. 107 (1999), no. 1, [] Appell, P. and Kapé de Fériet, J.: Fonctions hypergéoétriques et hypersphériques, Paris 196. [3] Ernst, T., The history of q-calculus and a new ethod, U. U. D. M. Report 000:16, ISSN , Departent of Matheatics, Uppsala University, 000. [4] Ernst, T, A new ethod and its Application to Generalized q-bessel Polynoials, U. U. D. M. Report 001:10, ISSN , Departent of Matheatics, Uppsala University, 001.

17 q-generating FUNCTIONS FOR ONE AND TWO VARIABLES. 17 [5] Exton H.: Multiple hypergeoetric functions and applications. Ellis Horwood, [6] Exton, H. q-hypergeoetric functions and applications, Ellis Horwood, [7] Exton, H., Two new ultivariable generating relations, Ark. Mat. 30 (199), no., [8] Gasper, G. and Rahan M., Basic hypergeoetric series, Cabridge, [9] Gelfand, I. M., Graev, M. I. and Retakh, V. S., General hypergeoetric systes of equations and series of hypergeoetric type, Russian Math. Surveys 47 (199), no. 4, [10] Jackson, F.H., On basic double hypergeoetric functions, Quart. J. Math., Oxford Ser. 13 (194), [11] Jackson, F.H., Basic double hypergeoetric functions, Quart. J. Math., Oxford Ser. 15(1944), [1] Hahn, W., Beiträge zur Theorie der Heineschen Reihen, Matheatische Nachrichten (1949), [13] Srivastava, H. M. and Manocha H. L., A treatise on generating functions, Ellis Horwood Series: John Wiley & Sons, Inc., New York, Departent of Matheatics, Uppsala University, P.O. Box 480, SE Uppsala, Sweden E-ail address: Thoas.Ernst@ath.uu.se