Some Results for Q-Functions of Many Variables

REND. SEM. MAT. UNIV. PADOVA, Vol. 112 (2004)

Some Results for q-Functions of Many Variables.

THOMAS ERNST (*)

ABSTRACT - We will give some new formulas and expansions for q-Appell-, q-Lau- ricella- and q-Kampé de Fériet functions. This is an area which has been explored by Jackson, R. P. Agarwal, W. A Al-Salam, Andrews, Kalnins, Mil- ler, Manocha, Jain, Srivastava, and Van der Jeugt. The multiple q-hypergeometric functions are defined by the q-shifted factorial and the tilde operator. By a method invented by the author [20]-[23], which also involves the Ward- Alsalam q-addition and the Jackson-Hahn q-addition, we are able to find q- analogues of corresponding formulas for the multiple hypergeometric case. In the process we give a new definition of q-hypergeometric series, illustrated by some examples, which elucidate the integration property of q-calculus, and helps to try to make a first systematic attempt to find summation- and reduction theorems for multiple basic series. The Jackson G q-function will be frequently used. Two definitions of a generalized q-Kampé de Fériet function, in the spirit of Karlsson and Srivastava [69], which are symmetric in the variables, and which allow q to be a vector are given. Various connections of transformation formulas for multiple q-hypergeometric series with Lie algebras and finite groups known from the literature are cited.

1. Historical introduction.

After Gauss’ introduction of the hypergeometric series in 1812, various versions of the multiple hypergeometric series (MHS) were developed by people whose names are now associated to the functions they introduced. In 1880, Paul Emile Appell (1855-1930) [7], [8] introduced so-

(*) Indirizzo dell’A.: Department of Mathematics, Uppsala University, P.O. Box 480, SE-751 06 Uppsala, Sweden. E-mail: thomasHmath.uu.se 200 Thomas Ernst me 2-variable hypergeometric series now called Appell functions

Q 8 (a)m 1m (b)m (b )m . 8 f 1 2 1 2 m1 m2 F1 (a; b, b ; c; x1 , x2 ) ! x1 x2 , 4 (1) m1 , m2 0 m ! m !(c) 1 / 1 2 m1 m2

´ max (Nx1N, Nx2N) E1.

Q 8 (a)m 1m (b)m (b )m . 8 8 f 1 2 1 2 m1 m2 F2 (a; b, b ; c, c ; x1 , x2 ) ! x1 x2 , (2) m1 , m2 40 m ! m !(c) (c 8) / 1 2 m1 m2

´ Nx1N1Nx2NE1.

Q 8 8 (a)m (a )m (b)m (b )m . 8 8 f 1 2 1 2 m1 m2 F3 (a, a ; b, b ; c; x1 , x2 ) ! x1 x2 , 4 (3) m1 , m2 0 m ! m !(c) 1 / 1 2 m1 m2

´ max (Nx1N, Nx2N) E1.

Q (a)m 1m (b)m 1m . 8 f 1 2 1 2 m1 m2 F4 (a; b; c, c ; x1 , x2 ) ! x1 x2 , (4) m1 , m2 40 m ! m !(c) (c 8) / 1 2 m1 m2

´ Nx1N1Nx2NE1.

REMARK 1. In the whole paper, the symbol f will denote definitions, except when we work with congruences. The four Lauricella functions of n variables were introduced in [50]. The first systematic treatment of this subject was written in French by J.M. Kampé de Fériet (1893-1982) and Appell [8]. The Kampé de Fériet functions, in their most general form, are defined in [p. 27] [69]. There are many generalizations of these functions in the literature, and we refer to the books Srivastava and Karlsson [69] and Exton [25]. Much of the research in MHS today is concerned with reduction formulas which reduce the degree of the function in some sense. As in the one-variable case, there are also confluent forms of MHS, which are obtained by using limits. We will now describe the method invented by the author [20]-[23], which also involves the Ward-Alsalam q-addition and the Jackson-Hahn q-addition. This method is a mixture of Heine 1846 [32] and Gasper- Rahman [26]. The advantages of this method have been summarized in [23, p. 495]. Some results for q-functions etc. 201

DEFINITION 1. The power function is defined by q a fe alog(q) . We always use the principal branch of the logarithm.

The variables a, b, c, a1 , a2 , R , b1 , b2 , RC denote parameters in hypergeometric series or q-hypergeometric series. The variables i, j, k, l, m, n, p, r will denote natural numbers except for certain cases where it will be clear from the context that i will denote the imagina- ry unit. The q-analogues of a complex number a and of the factorial function are defined by:

12q a (5) ]a(q f , qC0]1( , 12q

n ] ( ] ( ] ( (6) n q !f » k q , 0 q !f1, q C , k41

DEFINITION 2. In 1908 Jackson [39] reintroduced the Euler-Heine- Jackson q-difference operator

W(x)2W(qx) , if qC0]1(, xc0; . (12q)x ` dW (7) (Dq W)(x) f (x) if q41; / dx ` dW (0) if x40. ´ dx

If we want to indicate the variable which the q-difference operator is applied to, we write (Dq, x W)(x, y) for the operator.

REMARK 2. The definition (7) is more lucid than the one previously given, which was without the condition for x40. It leads to new so-called q-constants, or solutions to (Dq W)(x) 40.

DEFINITION 3. Let [16] e j denote the operator which maps f(xj ) to f(qj xj ). In our case all qj 4q. The following operator [41] will be useful in chapter 5.

f (8) u j xj Dq, xj . 202 Thomas Ernst

DEFINITION 4. Let the q-shifted factorial (compare [p. 38] [28]) be defined by

. 1, n40; f n21 (9) aa; qbn / » (12q a1m ) n41, 2, R , ´ m40

The Watson notation [26] will also be used

. 1, n40; f n21 (10) (a; q)n / » (12aq m ), n41, 2, R . ´ m40

Furthermore,

Q m (11) (a; q)Q f » (12aq ), 0 ENqNE1. m40

(a; q)Q (12) (a; q) f , acq 2m2a , m40, 1, R . a a (aq ; q)Q

aa; qbQ (13) aa; qba f , ac2m2a, m40, 1, R . aa1a; qbQ

C DEFINITION 5. In the following, will denote the space of complex Z 2pi numbers mod . This is isomorphic to the cylinder R3e 2piu , uR. log q The operator

A C C : O Z Z is defined by

pi (14) a O a1 . log q

Furthermore we define A A (15) aa; qbn f aa; qbn . Some results for q-functions etc. 203

By (14) it follows that

A n21 4 1 a1m (16) aa; qbn » (1 q ), m40 so that this time the tilde denotes an involution which changes the minus sign to a plus sign in all the n factors of aa; qbn . The following simple rules follow from (14). Clearly the first two equations are applicable to q-exponents. Compare [72, p. 110].

A A 2pi (17) a6bf a6b mod , log q

A A 2pi (18) a6b fa6b mod , log q

A (19) q a 42q a , where the second equation is a consequence of the fact that we work 2pi mod . log q

DEFINITION 6. Since products of q-shifted factorials occur so often, to simplify them we shall frequently use the more compact notation Let

(a) 4 (a1 , R , aA ) be a vector with A elements. Then

A (20) a(a); qbn f aa1 , R , aA ; qbn f » aaj ; qbn . j41

The following notation will be convenient.

(21) QE(x) fq x .

When there are several q:s, we generalize this

x (22) QE(x, qi ) fqi . The q-hypergeometric series was developed by Heine 1846 [32] as a generalization of the hypergeometric series.

DEFINITION 7. Generalizing Heine’s series, compare [26, p. 4], we shall define a q-hypergeometric series by × × × × (23) p1p 8f r1r 8 (a1 , R , ap ; b1 , R , brNq, zVs1 , R , sp 8 ; t1 , R , tr 8 ) f 204 Thomas Ernst

a× , R , a× s , R , s 8 f 1 pN V 1 p 4 p1p 8f r1r 8 y × × q, z z b1 , R , br t1 , R , tr 8 Q × × aa1 ; qbn Raap ; qbn (n) 11r1r 82p2p 8 4 ! k(21)n q 2 l 3 4 × × n 0 a1; qbn ab1 ; qbn Rabr ; qbn

p 8 r 8 3 n 21 z » (sk ; q)n » (tk ; q)n , k41 k41 where qc0 when p1p 8Dr1r 811, and

× . a, if no tilde is involved (24) a f / A ´ a otherwise .

REMARK 3. In a few cases, when 0 ENqNE1, the parameter a× in (23) will be the real plus infinity. × The factors aa; qbn then correspond to multiplication by 1. × × REMARK 4. The parameters ai and bi to the left of N in (23) are to be 2pi thought of as exponents; they are periodic mod . log q The general equation (23), with nonzero parameters to the left of V is used in certain special cases when we need factors (t; q)n in the q-series. One example is the q-analogue of a bilinear generating formula for La- guerre polynomials [23]. This notation will be further extended as follows.

EXAMPLE 1. Let Q 1 f k (25) Eq (x) ! x . 4 k 0 ]k(q ! We have

Q (2x 2 )k (11q k11 ) 2 2 42 (26) Dq Eq ( x ) x ! . 4 k 0 ]k(q ! This can’t be expressed in usual q-hypergeometric notation in a simple way. That’s why we suggest the following extension of the notation, where an extra k appears in the index to indicate the summation variable. This elucidates the integration property of q-calculus, i.e. to get the proper q-analogue, every factor in the hypergeometric formula has to be Some results for q-functions etc. 205 integrated in the (multiple) sum. The sum is the natural q-analogue of the integral, as is manifested by the definition of the q-integral

Q 2 11q k11 (27) D E (2x 2 ) 42x f Nq, 2x 2 (12q)V NNN . q q 1 0, k y2 2 2 z In general the new notation should look like, compare (72), (90).

DEFINITION 8.

× × » fi (k) a1 , R , ap s1 , R , sp 8 i (28) p1p 81p 9f r1r 81r 9, k Nq, zV NNN f y × R × R g (k)z b1 , , br t1 , , tr 8 » j j Q × × aa1 ; qbk R aap ; qbk (k) 11r1r 82p2p 8 f ! k(21)k q 2 l 3 4 × × k 0 a1; qbk ab1 ; qbk R abr ; qbk

8 p r 8 » fi (k) 3 k 21 i z »(sl ; q)k »(tl ; q)k , 4 4 l 1 l 1 » gj (k) j where qc0 when p1p 81p 9Dr1r 81r 911. We assume that the fi (k) × and gj (k) contain p 9 and r 9 factors of the form aa(k); qbk or (s(k); q)k respectively. In the same way as the G function plays a basic role in complex analy- sis, the G q function plays a fundamental role for q-calculus.

DEFINITION 9. Let the G q-function be defined by [71], [36], [47]

a1; qbQ . (12q)12x , if 0 ENqNE1 ax; qbQ (29) G (x) f q 21 / a1; q bQ (x) (q21)12x q 2 , if 1 Eq . 21 ´ ax; q bQ

The following notation for quotients of G q functions will sometimes be used.

DEFINITION 10. C R E a1 , , ap G q (a1 )RG q (ap ) (30) G ` ` f . q D F b1 , R , br G q (b1 )RG q (br )

We will give a longer exposition of the G q function in [24]. Basically 206 Thomas Ernst it has similar properties as the G function except for the simple 2kpi poles at x42n6 , n, kN. log q As was pointed out by Jackson [41, p. 78] the q-analogue of hypergeometric formulas with sums in the function arguments must contain a so- called q-addition. Jackson meant the Jackson-Hahn q-addition, compare [31, p. 362], which is given by

n n (k) y 6 nf 6 k 2 k n2k4 n Z 4 R (31) [x y]q !( ) q y x x ; q , n 0, 1, 2, . k40 g h g h k q x n The Ward-AlSalam q-addition was invented by Ward 1936 [76, p. 256] and Al-Salam 1959 [4, p. 240]

n n 5 n f k n2k 4 R (32) (a q b) ! a b , n 0, 1, 2, . k40g h k q Unlike the Ward-AlSalam q-addition, the Jackson-Hahn q-addition is neither commutative nor associative, but on the other hand, it can be written as a finite product. We will find the Ward-AlSalam q-addition more convenient for our purposes. The paper [11] was a breakthrough in the studies of expansions for MHS. By using an inverse pair of symbolic operators, Burchnall & Chaundy were able to prove a large number of expansion formulas for Appell functions. These results were later generalized by Verma [75] and Srivastava [64] to Kampé de Fériet functions. In 1942 Jackson [41] made the first attempt to find a q-analogue of [11]. Jackson [41, p. 70] and Agarwal [2, p. 187] defined four so-called normal q-Appell functions.

DEFINITION 11.

(33) F 1 (a; b, b 8; cNq; x1 , x2 ) f Q 8 aa; qbm1 1m2 ab; qbm1 ab ; qbm2 f m1 m2 ! x1 x2 . m , m 40 1 2 a1; qbm1 a1; qbm2 ac; qbm1 1m2

(34) F 2 (a; b, b 8; c, c 8Nq; x1 , x2 ) f Q 8 aa; qbm1 1m2 ab; qbm1 ab ; qbm2 f m1 m2 ! x1 x2 . m , m 40 8 1 2 a1; qbm1 a1; qbm2 ac; qbm1 ac ; qbm2 Some results for q-functions etc. 207

(35) F 3 (a, a 8; b, b 8; cNq; x1 , x2 ) f Q 8 8 aa; qbm1 aa ; qbm2 ab; qbm1 ab ; qbm2 f m1 m2 ! x1 x2 . m , m 40 1 2 a1; qbm1 a1; qbm2 ac; qbm1 1m2

(36) F 4(a;b;c,c 8Nq;x1,x2)f Q aa; qbm1 1m2 ab; qbm1 1m2 f m1 m2 ! x1 x2 , m , m 40 8 1 2 a1; qbm1 a1; qbm2 ac; qbm1 ac ; qbm2 with the same convergence regions as before, and with even sharper convergence when NqNE1 [41, p. 70]. k(m2) m2 2 For the four so-called abnormal q-Appell functions, x2 q replaces m2 x2 in the general term. We will see that normal functions (in a broader sense) correspond to the Ward-AlSalam q-addition, and abnormal functions, with k41, (in a broader sense) correspond to the Jackson-Hahn q-addition.

q-Contiguity relations for 2f 1 were found by Heine [33], and for q- Appell functions by Agarwal [3]. The first Saalschützian theorems for double series were published by Carlitz 1963 [13] and 1967 [14]. q-Analo- gues of [13] were found by W. A. Al-Salam [5]. W. N. Bailey (1893-1961) had been greatly influenced by Ramanujan as an undergraduate at Cam- bridge 1914 and wrote the first systematic treatment of hypergeometric series [10]. L. J. Slater attended Bailey’s lectures on q-hypergeometric series in 1947-50 at London University and wrote many important papers on this subject; among her pupils were Howard Exton. R. P. Agar- wal [2] visited Bailey, in 1953, and made the aforementioned contribu- tions to the subject. In 1966 Slater concluded [61, p. 234] that there seemed to be no systematic attempt to find summation theorems for basic Appell series, but Andrews [6] managed to prove some summation- and transformation-formulas for basic Appell series. Some of these do not have a hypergeometric counterpart. This is typical for q-calculus, we almost always get more equations than we had before. In a series of papers [52], [53] Miller has proved that by using the maximal Lie algebra generated by the first order differential recurrence relations satisfied by a MHS, we obtain addition theorems, transformations and generating functions. Then in 1980 Kalnins, Manocha and Mil- ler [45] developed a theory which enabled the powerful use of Lie alge- braic methods for the solutions of partial differential equations by two- variable hypergeometric series. As was shown in [1], this technique can 208 Thomas Ernst also be adapted to multiple q-hypergeometric series. This was the Lie algebra approach. There is also a finite group approach to transformation-formulas for q-MHS, which we will briefly summarize. In the footsteps of Rogers [57], [58], and Whipple [77], Lievens & Van der Jeugt [51] and Van der Jeugt [73] proved some formulas for q-MHS by using the invariance group or symmetry group of the particular double series. In 1996 Kuznetsov & Sklyanin [49] showed the connection between (multiple) Macdonald polynomials and q-Lauricella functions. Obviously we have not seen the end of this story yet as the subject gets more and more complex. Various q-analogues of Kampé de Fériet functions occur in the literature, compare [69, p. 349-350] and [70, p. 50]. We give a definition remini- scent of [26], which allows easy confluence to diminish the dimensions in (37) and (38), and has the advantage of being symmertic in the variables. Furthermore, q is allowed to be a vector and the full machinery of tilde operators and q-additions will be used. The first definition is a q-analogue of [69, (24), p. 38], in the spirit of Srivastava. The second definition is a q-analogue of [69, (24), p. 38] with the restraint [69, (29), p. 38], due to Karlsson. It will be clear from the context which of the definitions we use.

DEFINITION 12. The notation ! denotes a multiple summation with m the indices m1 , R , mn running over all non-negative integer values. In n this connection we put mf ! mj . j41

DEFINITION 13. Let

(a), (b), (gi ), (hi ), (a 8), (b 8), (gi8), (hi8) have dimensions

A, B, Gi , Hi , A 8, B 8, Gi8, Hi8. Let

11B1B 81Hi 1Hi82A2A 82Gi 2Gi8F0, i41, R , n. Then the generalized q-Kampé de Fériet function is defined by

(a×): (g× ); R; (g× ); (a 8): (g 8); R; (g 8); A1A 8: G11G18; R; Gn1Gn8 1 n 1 n (37) F 1 8 1 8 1 8 Nq; xV f B B : H1 H1 ; R; Hn Hn y × × × 8 8 8 z (b): (h1 ); R; (hn ); (b ): (h1 ); R; (hn ); Some results for q-functions etc. 209

n × 8 × 8 mj a(a); q0 bm 1R1m ( (a ); q0 )m 1R1m » (a(gj ); qj bm ( (gj ); qj )m xj ) 1 n 1 n 4 j j f j 1 3 ! n m × 8 × 8 a(b); q0 bm 1R1m ( (b ); q0 )m 1R1m » (a(hj ); qj bm ( (hj ); qj )m a1; qj bm ) 1 n 1 n j j j j41

n 1 1 82 2 81 1 82 2 8 3(21)!j41 mj (1 Hj Hj Gj Gj B B A A ) 3

n m mj 1 82 2 8 1 1 82 2 8 QE (B B A A ) , q0 » QE (1 Hj Hj Gj Gj ) , qj . g g 2 h h j41 u u 2 v v

DEFINITION 14. Let

(a), (b), (gi ), (hi ), (a 8), (b 8), (gi8), (hi8) have dimensions A, B, G, H, A 8, B 8, G 8, H 8, (i .

Let 11B1B 81H1H 82A2A 82G2G 8F0. Then the generalized q-Kampé de Fériet function is defined by

(a×): (g× ); R; (g× ); (a 8): (g 8); R; (g 8); A1A 8: G1G 8 1 n 1 n (38) F 1 8 1 8 Nq; xV f B B : H H y × × × 8 8 8 z (b): (h1 ); R; (hn ); (b ): (h1 ); R; (hn );

n 1 1 82 2 81 1 82 2 8 3(21)!j41 mj (1 H H G G B B A A ) 3

n m mj 3 1 82 2 8 1 1 82 2 8 QE (B B A A ) , q0 » QE (1 H H G G ) , qj . g g 2 h h j41 u u 2 v v

REMARK 5. In case we want to get rid of the Q symbol, we just take away all the 21 and q-powers in the definitions. In the one-variable case, this was the original approach, see Thomae 1871 [72, p. 111], [62]. How- ever in this case we have to watch out for convergence and we lose the confluence property. 210 Thomas Ernst

2. Transformations for basic double series.

In some of the proofs we will need the following q-summation formulas. A q-analogue of the Dixon-Schafheitlin theorem [19], [59, p. 24 (22)] is a corollary of Jackson’s 8f 7 sum. A 1 a, b, c,11 a a 2 11 2b2c N 2 4 (39) 4f 3 A q, q y 1 z 11a2b, 11a2c, a 2

a a 11a2b, 11a2c, 11 , 11 2b2c 2 2 4G q . y 1 1 2 2 1 a 2 1 a 2 z 1 a, 1 a b c, 1 b, 1 c 2 2

A q-analogue of Kummer’s formula [48], the Bailey-Daum summation formula, was proved independently by Bailey 1941 [9] and Daum 1942 [18]. This was the first example of a q-analogue of a summation formula for a hypergeometric series with argument 21.

THEOREM 2.1.

12b (40) 2f 1 (a, b; 11a2bNq, 2q ) 4 A a a A 11a2b, 11 a1 1 2b,1; qbQ 4 2 2 G q A . y 1 1 a 2 z a A 1 a, 1 b a11 ,12b; qbQ 2 2 where Nq a112b NE1 and 11a2bc0, 21, 22 R. A a PROOF. Put c4 in the previous formula or use [26, (1.8.1) p. 14]. 2 The idea to use the q-Dixon-Schafheitlin theorem for this proof was mentioned in Daum’s thesis [17], chapter 3. r

If a is a negative integer, (40) may be reformulated in the form Some results for q-functions etc. 211

THEOREM 2.2.

12b (41) 2f 1 (22N, b; 122N2bNq, 2q ) f

2N 2N (k)1k(122N2b) ab; qb f ! q 2 k f k40g h 2 2 k q a1 2N b; qbk

2 N 1 2N ab; qb 2N ab; qb f2 ! (21)k k 1 (21)N N 4 k40 g h 1 2 g h 1 k q a2N b k; qbk N q aN b; qbN A ab; qb 1 4 N ; q 2 . 1 aN b; qbN o 2 pN

12b (42) 2f 1 (2N, b; 12N2bNq, 2q ) 40, N odd.

PROOF. The validity has been confirmed by Axel Riese (RISC) in Linz using Mathematica. r

REMARK 6. For b41Q this is a result of Gauss [27], [29]. Equa- tion (41) is a q-analogue of [55, p. 43].

The following corollary enables us to find a relation for G q-functions with negative integer argument.

COROLLARY 2.3. A A C122N2b, 12NE 12N, b; q 1 4 a bN 2 (43) G q ` ` A ; q . D122N, 12N2bF 2 2 1 o 2 pN a1 N b, b N; qbN

PROOF. Just equate (40) and (41). r

We continue with a few examples where the Jackson [40], [26, p. 11, 1.5.4] q-analogue of the Euler-Pfaff-Kummer transformation is used.

1 b a (44) 2f 1 (a, b; cNq, t) 4 2f 2 (a, c2b; cNq, tq V2; tq ). (t; q)a

When no convergence region is given for a certain formula, we first assume that the absolute value of the function arguments are small.

The following three expansions of F 1 in terms of one-variable f are 212 Thomas Ernst q-analogues of the Appell and Kampé de Fériet [8, p. 24 (27), (278), (279)] formulas.

Q aa, b; qb x m1 8 N 4 m1 1 3 (45) F 1 (a; b, b ; g q; x1 , x2 ) ! m 40 1 a1, g; qbm1 (x2 ; q)a1m1

b8 a1m1 3 2f 2 (a1m1 , g2b81m1 ; g1m1 Nq, x2 q V2; x2 q ).

Q aa, b; qb x m1 8 N 4 m1 1 3 (46) F 1 (a; b, b ; g q; x1 , x2 ) ! m 40 1 a1, g; qbm1 (x2 ; q)b8 1 8 3 8 2 1 N a m1 2 b 2f 2 (b , g a; g m1 q, x2 q V ; x2 q ).

Q aa, b; qb x m1 8 N 4 m1 1 3 (47) F 1 (a; b, b ; g q; x1 , x2 ) ! m 40 1 a1, g; qbm1 (x2 ; q)a1b82g a1b82g 3 2f 1 (g2a, g2b81m1 ; g1m1 Nq, x2 q ). B y J a c k s o n [ 4 0 ] w e o b t a i n t h e f o l l o w i n g t w o e x p a n s i o n s o f

2f 1 (Nq, x1 5q x2 ):

Q aa, b; qb x m1 N 5 4 m1 1 3 (48) 2f 1 (a, b; g q, x1 q x2 ) ! m 40 1 a1, g; qbm1 (x2 ; q)a1m1 1 1 3 1 2 1 N b m1 2 a m1 2f 2 (a m1 , g b; g m1 q, x2 q V ; x2 q ).

Q aa, b; qb x m1 N 5 4 m1 1 3 (49) 2f 1 (a, b; g q, x1 q x2 ) ! m 40 1 a1, g; qbm1 (x2 ; q)b1m1 1 1 3 1 2 1 N a m1 2 b m1 2f 2 (b m1 , g a; g m1 q, x2 q V ; x2 q ). By [34, p. 115] we obtain

Q aa, b; qb x m1 N 5 4 m1 1 3 (50) 2f 1 (a, b; g q, x1 q x2 ) ! m 40 1 a1, g; qbm1 (x2 ; q)a1b2g1m1

a1b2g1m1 3 2f 1 (g2a, g2b; g1m1 Nq, x2 q ).

The following three expansions of F 2 in terms of one-variable f are q- analogues of [8, p. 24 (30), (308), (309)]

Q aa, b; qb x m1 8 8N 4 m1 1 3 (51) F 2 (a; b, b ; g, g q; x1 , x2 ) ! m 40 1 a1, g; qbm1 (x2 ; q)a1m1

b8 a1m1 3 2f 2 (a1m1 , g82b8; g8Nq, x2 q V2; x2 q ). Some results for q-functions etc. 213

Q aa, b; qb x m1 8 8N 4 m1 1 3 (52) F 2 (a; b, b ; g, g q; x1 , x2 ) ! m 40 1 a1, g; qbm1 (x2 ; q)b8 1 8 3 8 82 2 8N a m1 2 b 2f 2 (b , g a m1 ; g q, x2 q V ; x2 q ).

Q aa, b; qb x m1 8 8N 4 m1 1 3 (53) F 2 (a; b, b ; g, g q; x1 , x2 ) ! m 40 1 a1, g; qbm1 (x2 ; q)a1b82g81m1

a1b82g81m1 3 2f 1 (g82a2m1 , g82b8; g8Nq, x2 q ). By [40], we get a q-analogue of [8, p. 25 (31)]

Q 1 aa, b; qb x m1 8 8N 4 m1 1 a 3 F 2 (a; b, b ; g, g q; x1 , x2 ) ! (x2 q ; q)m1 m 40 (x2 ; q)a 1 a1, g; qbm1

Q b8 m2 1 82 8 m aa m1 , g b ; qbm2 (x2 q ) ( 2) 3 2 m2 2 4 ! 1 ( 1) q a m1 m2 40 8 a1, g ; qbm2 (x2 q ; q)m2

1 a : b; g82b8; 2: 2; 4 F 1: 1; 1 Nq; x , x q b8 V . 1: 1; 2y 1 2 a z (x2 ; q)a 2: g; g8, Q; x2 q : 2;

The following three expansions of F 3 in terms of one-variable f are q- analogues of [8, p. 25 (33), (338), (339)]

Q aa, b; qb x m1 8 8 N 4 m1 1 3 (54) F 3 (a, a ; b, b ; g q; x1 , x2 ) ! m 40 1 a1, g; qbm1 (x2 ; q)a8 b8 a8 3 2f 2 (a8, g2b81m1 ; g1m1 Nq, x2 q V2; x2 q ).

Q aa, b; qb x m1 8 8 N 4 m1 1 3 (55) F 3 (a, a ; b, b ; g q; x1 , x2 ) ! m 40 1 a1, g; qbm1 (x2 ; q)b8 a8 b8 3 2f 2 (b8, g2a81m1 ; g1m1 Nq, x2 q V2; x2 q ).

Q aa, b; qb x m1 8 8 N 4 m1 1 3 (56) F 3 (a, a ; b, b ; g q; x1 , x2 ) ! m 40 1 a1, g; qbm1 (x2 ; q)a81b82g2m1

a81b82g2m1 3 2f 1 (g2a81m1 , g2b81m1 ; g1m1 Nq, x2 q ). Our next task is to find q-analogues of some general formulas of Car- lson [15]. We first recall the following two equivalent forms of the q-Van- dermonde theorem. The second one also appeared in [41, (29) p. 74] and [42, (60) p. 56]. 214 Thomas Ernst

LEMMA 2.4.

ab; qb ab 8; qb ab1b 8; qb (57) ! m n q 2nb 8 4q 2Nb 8 N . 1 4 m n N a1; qbm a1; qbn a1; qbN

ab; qb ab 8; qb ab1b 8; qb (58) ! m n q nb 4 N . 1 4 m n N a1; qbm a1; qbn a1; qbN

The order classification [35] for double hypergeometric series (DHS) was prevalent for a long time. Carlson [15] showed evidence in support of the need for a new classification of DHS.

THEOREM 2.5. A q-analogue of Carlson’s transformation [15, (8), p. 223].

b;q b 8;q a bma bn 2nb 8 5 m n4 (59) ! q (x1 qx2) x2 1 4 m n N a1;qbma1;qbn

b; q 4 1 8 a bm 2nb 8 m n ab b ; qbN ! q x1 x2 . 1 4 m n N ab1b 8, 1; qbm a1; qbn

PROOF.

N N2n b; q 2 b 8; q N2n 4 a bN n a bn 2nb 8 m 8 N2m 8 4 (60) LHS ! q ! x1 x2 n40 2 m 840g 8 h a1; qbN n a1; qbn m q

N m N2m N2m x x ab; qb 2 ab 8; qb by(57) 4 ! 1 2 ! N n n q 2nb 8 4 4 4 4 m 0 a1; qbm n 0 a1; qbn a1; qbN2n2m

N m N2m x x ab; qb ab1b 81m; qb 2 4 ! 1 2 q 2b 8(N2m) m N m 4 4 m 0 a1; qbm a1; qbN2m

m n x x ab; qb ab1b 81m; qb 2 4 ! 1 2 q 2b 8 n m N m 4RHS. 1 4 m n N a1; qbm a1; qbn

REMARK 7. One obtains three similar results by i) using equation m m (58) in place of (57), or ii) using [x2 1x1 ]q in place of (x1 5q x2 ) or iii) ma- king both of these changes. Some results for q-functions etc. 215

COROLLARY 2.6. A q-analogue of [15, (4), p. 222], which is valid when

Nx1N1Nx2NE1, acc, ccb1b 8, bc1, b 8c1, neither c nor b1b 840 or 2N.

Q a; q 1 b; q b 8; q a bm n a bm a bn 2nb 8 5 m n 4 (61) ! q (x1 q x2 ) x2 4 m, n 0 ac; qbm1n a1; qbm a1; qbn

Q a, b1b 8; q 1 b; q 4 a bm n a bm 2nb 8 m n ! q x1 x2 . 4 m, n 0 ac; qbm1n ab1b 8, 1; qbm a1; qbn

REMARK 8. Three other q-analogues of Carlson’s transformation may be obtained via the three variants of Theorem 2.5 mentioned above.

3. q-Analogues of some of Srivastava’s formulas.

In this chapter we will derive some more complex q-analogues than before. When certain special cases of these (non-q)-formulas have been published, we try to find q-analogues of them too. We will prove two general multiple formulas via the q-Vandermonde sums, and two general formulas by use of (41). The Bailey-Daum theorem will also be used. We first derive two q-analogues of the general formula [65, (4), p. 295] (62) and (66).

Q THEOREM 3.1. If ]Cn (n40 is a sequence of arbitrary complex numbers then

n 1 Q m1n 2(2) sn Q N C 1 x q C x m n 4 N 2 1 1 (62) ! ! as 1 n N; qbN . 4 4 m, n 0 a1, n; qbm a1, s; qbn N 0 a1, n, s; qbN

PROOF. (63) LHS4

Q C x N N a2n112N, 2N; qb 4 ! N ! n q n(211n12N1s) 4RHS. r 4 4 N 0 a1, n; qbN n 0 a1, s; qbn 216 Thomas Ernst

1 COROLLARY 3.2. A first q-analogue of [65, (8), p. 296]. If NxNE 4 then

n 1 Q m1n 2(2) sn al, m; qb 1 x q (64) ! m n 4 4 m, n 0 an, 1; qbm as, 1; qbn A A 1 2 1 2 1 1 n s 1 n s 1 n s n s n Q al, m, , , , ; qbn x 2 2 2 2 4 ! . 4 n 0 an, 1, s, n1s21; qbn

REMARK 9. The convergence radius R(q) in Corollary 3.2 is the one for q41. We feel that the convergence radius for NqNE1 is in fact big- ger, and that R(q) E1. This is justified by the following computation.

Denote the terms of the RHS sum in (64) an (x). Then the quotient crite- rion gives

a 1 (x) (65) lim nK1QN n 1 N 4 lim nK1Q3 an (x) A A 1 2 1 2 1 1 1 1 n s 1 1 n s 1 1 n s 1 n s 1 al n, m n, n, n, n, n; qb1 2 2 2 2 3 N xN4NxN. an1n,11n, s1n, n1s211n; qb1 This implies that the RHS of (64) should converge for NxNE1, when NqNE1. However, things are not so easy as we have an equality between a double sum and a simple sum as functions of the two complex variables x and q. In general, the convergence works best when both NqN and NxN are far away from 1. The same reasoning obtains for Corollary 3.4 and Corollary 3.6.

Q THEOREM 3.3. If ]Cn (n40 is a sequence of arbitrary complex numbers then

n 1 2 1 2 1 Q m1n 2(2) n( 2(m n) n 2) C 1 x q (66) ! m n 4 4 m, n 0 a1, n; qbm a1, s; qbn

Q C x N 4 N 2 1 1 N(12n2N) ! as 1 n N; qbN q . 4 N 0 a1, n, s; qbN Some results for q-functions etc. 217

COROLLARY 3.4. A second q-analogue of [65, (8), p. 296].

n 1 2 1 2 1 Q m1n 2(2) n( 2(m n) n 2) al, m; qb 1 x q (67) ! m n 4 4 m, n 0 an, 1; qbm as, 1; qbn A A 1 2 1 2 1 1 n s 1 n s 1 n s n s n n(12n2n) Q al, m, , , , ; qbn x q 2 2 2 2 1 4 ! , NxNE . 4 n 0 an, 1, s, n1s21; qbn 4

THEOREM 3.5. A q-analogue of [65, (5), p. 295]. Q If ]Cn (n40 is a sequence of arbitrary complex numbers then

Q n m1n 2mn (21) C 1 x q (68) ! m n 4 4 m, n 0 a1, s; qbm a1, s; qbn

A (n)1ns Q 2n 2 1 2 2 n C2n x a s 1 2n; qbn q 4 ! (21) A . n40 s; q a b2n a1, 1, s; qbn

PROOF.

Q C x N N a2s112N, 2N; qb (69) LHS4 ! N ! (21)n n q n(N1s) 4 4 4 N 0 a1, s; qbN n 0 a1, s; qbn A Q N 2 1 2 C x a s 1 N; qbN/2 1 4 ! N ; q 2 4 4 N N 0, N even a1, s; qbN 2 1 2 o 2 pN/2 a s 1 ; qbN/2 2

Q A C x 2n a2s 112 2n; qb 1 4 ! 2n n ; q 2 4RHS . r 4 2 1 2 n 0 a1, s; qb2n a s 1 n; qbn o 2 pn

1 COROLLARY 3.6. A q-analogue of [65, (9), p. 296]. If NxNE then 4

Q m1n 2mn al, m; qb 1 x q (70) ! (21)n m n 4 4 m, n 0 as, 1; qbm as, 1; qbn 218 Thomas Ernst

A A A A l l11 m m11 l l11 m m11 , , , , , , , ; q x 2n Q 2 2 2 2 2 2 2 2 » «n 4 ! n40 A A 3 s s11 s s11 , , , , s, 1; q 2 2 2 2 » «n A 2 1 2 n 1 a s 1 2n; qbn n (2) ns 3 A (21) q . a1; qbn

By (62) we get

COROLLARY 3.7. A first q-analogue of a reduction formula for the Humbert function [65, (12), p. 296].

Q Q 2 1: 2; 0 m : , ; ; s (71) F 1: 1; 1 Nq; x, xq 4 y Q: n; s; z A A n1s21 n1s n1s21 n1s 4 f m, , , , Nq, x . 5 4 y 2 2 2 2 z n, s, n1s21, Q

By (68) we get

COROLLARY 3.8. A q-analogue of a reduction formula for the Hum- bert function [65, (13), p. 296].

Q 1 2 (21)n am; qb x m n q mn (72) ! n 4 4 m, n 0 a1, n; qbm a1, n; qbn

A A m m11 m m11 , , , ; q Q A (n)1nn 2 n 2 1 2 2n 2 2 2 2 2 ( 1) a n 1 2n; qbn x q » «n 4 ! A n40 A A f a1, 1; qbn n n11 n n11 , , , ; q 2 2 2 2 » «n Some results for q-functions etc. 219 A 1 A 1 m m 1 m m 1 Q , , , , 2 A1 2 f 2 2 2 2 N 2 2 n V2NNNa n 1 2k; qbk 6f 6, k A q, x q 2 . y 1 A 1 A z n n 1 n n 1 , , , , n,1 2 2 2 2

COROLLARY 3.9. A q-analogue of [65, (14), p. 297], which leads to a relation for the product of the two Jackson q-Bessel functions [26], [37], [38].

s (73) 2f 1 (Q, Q; nNq, x) 0f 1 (2; sNq, xq ) 4 A A n1s21 n1s n1s21 n1s 4 f , , , Nq, x . 4 3 y 2 2 2 2 z n, s, n1s21

THEOREM 3.10. The first q-analogue of [65, (16), p. 297].

Q m1n 2ns Q n C 1 x an; qb as; qb q C x an1s; qb (74) ! m n m n 4 ! n n q 2ns . 4 4 m, n 0 a1; qbm a1; qbn n 0 a1; qbn

PROOF.

Q 2 1 2 N C x N a2N, s; qb an; qb q n( n 1 s) (75) LHS4 ! ! N n N 4 4 4 N 0 n 0 a1; qbN a1, 2n112N; qbn

Q C x N an, 2n112s2N; qb 4 ! N N 4RHS . r 4 N 0 a1, 2n112N; qbN There is also a second q-analogue of [65, (16), p. 297].

THEOREM 3.11.

Q m1n nn Q n C 1 x an; qb as; qb q C x an1s; qb (76) ! m n m n 4 ! n n . 4 4 m, n 0 a1; qbm a1; qbn n 0 a1; qbn By putting

aa; qbn c2a2b (77) Cn 4 , n4b, s4b 8, x4q ac; qbn 220 Thomas Ernst in the two equations above, we get the following two special cases, which are q-analogues of [8, (24), p. 22].

COROLLARY 3.12.

c2a2b c2a2b2b 8 (78) F 1 (a; b, b 8; cNq; q , q ) 4

C 2 2 2 8E c2a2b2b 8 c2a2b 8 c, c a b b 4F 1 (a; b, b 8; cNq; q , q ) 4G q ` ` , Dc2a, c2b2b 8F where Nq c2a2b2b 8NE1.

For n42, the above theorem implies the following reduction theorem for a q-Lauricella function from [21, p. 5 (27)].

THEOREM 3.13.

(n) 2b2 2b2 2b3 2b2 2R2bn (79) F D (a, b1 , R , bn ; cNq; x, xq , xq , R , xq ) 4

2b2 2R2bn 42f 1 (a, b1 1R1bn ; cNq, xq ), where

(n) (80) F D (a, b1 , R , bn ; cNq; x1 , R , xn ) f

n mj aa; qbm 1R1m » abj ; qbm xj 1 n 4 j f j 1 N N R N N E ! , max ( x , , xn ) 1. m n ac; qbm 1R1m » a1; qbm 1 n j j41

THEOREM 3.14. A q-analogue of [65, (17), p. 297].

Q 2 n m1n Q 2n A ( 1) Cm1n x av; qbm av; qbn C x av, v; qb (81) ! 4 ! 2n n . 4 4 2 m, n 0 a1; qbm a1; qbn n 0 a1; q bn

PROOF. See the proof of (88). r

By applying the Bailey-Daum theorem, we can prove the following Some results for q-functions etc. 221

THEOREM 3.15. A q-analogue of [63, p. 104]. If ac negative integer then

12b 8 12b 8 (82) F 1 (a; b, b 8; 11a1b2b 8Nq; q , 2q ) 4 A a A a 1 1 2 b 8,1; q 11a1b2b 8, 12b 8, 11 2 2 » «Q , 4G q y 1 1 2 8 1 a 2 8z A 1 a, 1 b b , 1 b a A 2 11 ,12b 8 ; q 2 » «Q where Nq 12b 8NE1.

PROOF.

Q aa, b 8; qb (83) LHS4 ! n (21)n q n(12b 8) 4 4 n 0 a1, 11a1b2b 8; qbn

Q aa1n, b; qb 4 ! m q m(12b 8) 4 4 m 0 a1, 11a1b2b 81n; qbm

Q a, b 8; q 11a1b2b 8, 12b 8 4 a bn 2 n n(12b 8) 2 2n4 ! ( 1) q G q (1 q) 4 y z n 0 a1; qbn 11b2b 8, 11a2b 81n

Q a, b 8; q 1 1 2 8 2 8 4 a bn 2 n n(12b 8) 1 a b b , 1 b 4 ! ( 1) q G q 4 1 2 8 k 1 2 8 1 2 8l n 0 a1, 1 a b ; qbn 1 b b , 1 a b

4RHS . r When a is a negative integer, the above theorem may be reformulated as

THEOREM 3.16.

12b 8 12b 8 (84) F 1 (22N; b, b 8; 11a1b2b 8Nq; q , 2q ) 4 A 122N1b2b 8, 12b 8 ab 8; qb 1 4G N ; q 2 . q k 2 2 8 1 2 8l 1 8 1 2N b , 1 b b aN b ; qbN o 2 pN

12b 8 12b 8 (85) F 1 (2N; b, b 8; 11a1b2b 8Nq; q , 2q ) 40, N odd . The following theorem is a q-analogue of Srivastavas generalization 222 Thomas Ernst of Carlson’s identity from [67, (3), p. 139]. Notice the difference in character to the equations in chapter two.

Q THEOREM 3.17. If ]Cn (n40 is a bounded sequence of complex numbers,

x1 c0, vc1, sc1, v1sc0, 21, 22R and the two double series are absolutely convergent, then

m1n 2 x1 2s Q Cm1n x2 q ; q av; qbm as; qbn g x h (86) ! 2 m 3 4 m, n 0 a1; qbm a1; qbn m n 3QE 2 1 1sm1n(12m2n) 4 g g 2 h g2h h

Q m n C 1 x x av; qb av1s1m; qb n 4 ! m n 1 2 m n QE 2 2mn . m, n40 g g h h a1; qbm a1; qbn 2

PROOF. We will use the q-Euler-Pfaff equation (44). (87) RHS4

n 2 n N x2 2 2 2 2 Q N ( 1) CN x1 av; qbN a N, 1 v s N; qbn g x h 4 ! ! 1 q n(11s) 4 4 4 N 0 n 0 a1; qbN a1, 12v2N; qbn

Q 2 C x N v; q s,2N x 4 N 1 a bN N 2 2 22v2N V 4 ! 2f 2 q, q 2 x2 11s2N N40 x y 2 2 x q z 1 1 v N 2 2 1 s 1 x a1; qbN g q ; qh 1 x1 2N

n n x2 22v2N (2) Q N q q C x N v; q 2N, s; q g x h 4 ! N 1 a bN ! a bn 1 4 N40 1; q n40 1, 12v2N; q x2 1 a bN a bn g2 q 1 s ; qh x1 n2N

Q N C x N av; qb a2N, s; qb y n 4 ! N 1 N ! n q 22v2N 3 4 4 2 2 g h N 0 a1; qbN n 0 a1, 1 v N; qbn x1

2 x N2n n x N n 3 2 1 q 2s ; q QE 2 1 1s(N2n) 2 4 g x h 2 g g 2 h g2h hg h 2 N n x1 Some results for q-functions etc. 223

Q N N C x as; qb av; qb 2 x 4 ! ! N 1 n N n 2 1 q 2s ; q 3 4 4 g h N 0 n 0 a1; qbn a1; qbN2n x2 N2n

N2n n x N 3QE 2 1 1s(N2n)1n(12N) 2 4LHS. r g g 2 h g2h hg h x1

4. q-Analogues of reducibility theorems of Karlsson.

Karlsson [46] has derived two interesting reduction formulas of general character for triple sums, which can be used to deduce many reduction formulas for hypergeometric functions in three variables [69, p. 311]. The formulas (39) and (41) are used in the proofs. Clearly many more reducibility formulas of this type can be deduced. We illustrate with a q-analogue of one interesting special case known from the literature.

THEOREM 4.1. A q-analogue of [46, p. 200]. Q If ]Cm, n (m, n40 is a sequence of arbitrary complex numbers then

Q m n 2 p Cm, n1p ab; qbn ab; qbp x1 x2 ( x2 ) (88) ! 4 4 m, n, p 0 a1; qbm a1; qbn a1; qbp

Q Q A m 2k Cm, 2k ab, b; qbk x1 x2 4 ! ! A . m40 k40 a1; qbm a1, 1; qbk

PROOF.

Q t 2 m t 2 p p(12b) Cm, t ab; qbt ab, t; qbp x1 x2 ( 1) q (89) LHS4 ! ! 4 4 4 m, t 0 p 0 a1; qbm a1; qbt a1, 12t2b; qbp A 1 2 Q Q m t ab; qbt/2 ; q Cm, t ab; qbt x1 x2 o 2 pt/2 4 ! ! 4RHS. r m40 t40, t even 1; q 1; q t a bm a bt b1 ; q o 2 pt/2

REMARK 10. We obtain (81) by putting x1 40 in the theorem above. 224 Thomas Ernst

COROLLARY 4.2. A q-analogue of [46, 2.5, p. 201].

a c, (3) c2a 2 (90) F D (a11, 11a2c, b, b; cNq; 2q , x, 2x)4G q 3 y 2 a z a, c 2

A a A b, b, a 2 2 1c1k; q 3 f Nq, x 2 V2NNN 2 . 3 2, k A a » « 1 2 y1, c2 a 1 c z a 2 1k A2 1 2 a c a ; qba112c (1 q )

PROOF. Put

a11; q 1 11a2c; q 4 a bm n a bm 42 c2a Cm, n ; x1 q in (88) . ac; qbm1n (91) RHS4 A Q 1 2 1 aa; qb2k a1 a c, a 2k; qbm ab, b; qbk 2k m m(c2a) 4 ! A x (21) q 4 m, k40 c; q 1, c12k; q a b2k a bm a1, 1; qbk

a Q A C c12k, 111k E a; q b, b; q 4 a b2k a bk 2k 2 3 ! A x G q ` ` k40 c; q a a b2k a1, 1; qbk 11a12k, c2 1k D 2 F A a A 2 1c1k,1; q A 2 a » «Q Q , b, b; q 3 o 2 pk a A A 4 ! x 2k 3 111 k, c2a ; q k40 a A o 2 pQ c2 ,1, 1; q o 2 pk A a a 2 1c1k; q C c, E 2 » «a112c 3G 2 q ` ` a 4LHS. r 2 a A 1k a, c a c2a ; qb 1 2 (11q 2 ) D 2 F a 1 c

THEOREM 4.3. A q-analogue of [46, 4.1 p. 202]. Some results for q-functions etc. 225

Q If ]Cm, n (m, n40 is a sequence of arbitrary complex numbers then

A p n1p (n21) 2 m n 2 p 2 Cm, n1p aa, b; qbn a, b,1 ; q x1 x2 ( x2) q Q 2 » «p 4 (92) ! m, n, p40 A n1p a1; qbm a1; qbn 1, 2 ; q 2 » «p

A A Q 2 1 m 2k 2kb Cm, 2k aa, b,1 k, b; qbk aa b; qb2k x1 x2 q 4 ! . m, k40 A A a1; qbm a1, a1b, 1 2b2 k,1; qbk

PROOF. Compare [56, p. 56].

Q n (93) ! ! 3 m, n40 p40

A p n (p)1p(22n2a2b)1 (n2p21) 2 2 m n 2 2 Cm, n aa, b; qbn a, b, n,1 ; q x1 x2 q 2 » «p 3 by(39) A 4 n a1;qbma1;qbn 1,12n2b,12n2a,2 ;q » 2 «p

C 2 2 2 2 2 n 2 2 2 n E Q C aa, b; qb x m x n 1 b n,1 a n,1 ,1 a b 4 m, n n 1 2 2 2 4 ! G q 4 ` n n ` m, n 0 a1; qbm a1; qbn 12n,12a2b2n,12b2 ,12a2 D 2 2 F

C 2 2 2 2 2 n E Q Q C aa, b; qb x m x n 1 a n, 1 a b 4 m, n n 1 2 2 3 ! ! G q 4 4 ` n ` m 0 n 0, n even a1; qbm a1; qbn 12a2b2n, 12a2 D 2 F 226 Thomas Ernst A n A 12 , b; q 2 » «n/2 Q m 2k 3 1 Cm, 2k aa, b; qb2k x1 x2 A ; q 2 4 ! 3 n n 4 1 2 2 b, b1 ; q o 2 pn/2 m, k 0 a1; qbm a1; qb2k 2 2 » «n/2

A A a12a2b22k,12k , b; qb 3 k 1 A ; q 2 4RHS. r 12a22k, 1 2k2 b, b1k; q a bk o 2 pk

5. q-Analogues of Burchnall-Chaundy expansions.

This chapter is mainly based on the following inverse pair of symbolic operators defined in [11]. We will get an improved version of [41] in the process, compare [2]. The operators u 1 and u 2 from (8), chapter 1 will be extensively used.

DEFINITION 15.

C 1 1 E C 1 1 E h, h u 1 u 2 h u 1 , h u 2 (94) ˜q (h) fG q ` ` , D (h) fG ` ` . D F q q D F h1u 1 , h1u 2 h1u 1 1u 2 , h

Then

m n m n (95) ˜q (h)ah; qbm ah; qbn x1 x2 4 ah; qbm1n x1 x2 .

Recall that

(96) 2f 1 (a, b; cNq, x1 ) 2f 1 (a 8, b 8; c 8Nq, x2 ) 4 Q a, b; q a 8, b 8; q 4 a bm a bn m n ! x1 x2 . 4 m, n 0 a1, c; qbm a1, c 8; qbn

Q a; q 1 b; q 1 N 5 4 a bm n a bm n m n (97) 2f 1 (a, b; c q, x1 q x2 ) ! x1 x2 . 4 m, n 0 a1; qbm a1; qbn ac; qbm1n

We can write symbolically (the last three equations first appeared in [41, Some results for q-functions etc. 227

(14) p. 72])

(98) F 2 (a; b, b 8; c, c 8Nq; x1 , x2 ) 4

4˜q (a)2 f 1 (a, b; cNq, x1 )2 f 1 (a, b 8; c 8Nq, x2 ).

(99) F 3 (a, a 8; b, b 8; cNq; x1 , x2 ) 4

4D q (c) 2f 1 (a, b; cNq, x1 ) 2f 1 (a 8, b 8; cNq, x2 ).

(100) F 1 (a; b, b 8; cNq; x1 , x2 ) 4

4˜q (a) D q (c)2 f 1 (a, b; cNq, x1 )2 f 1 (a, b 8; cNq, x2 ).

(101) F 1 (a; b, b 8; cNq; x1 , x2 ) 4˜q (a) F 3 (a, a; b, b 8; cNq; x1 , x2 ).

(102) F 1 (a; b, b 8; cNq; x1 , x2 ) 4D q (c) F 2 (a; b, b 8; c, cNq; x1 , x2 ).

(103) F 4 (a, b; c, c 8Nq; x1 , x2 ) 4˜q (b) F 2 (a; b, b; c, c 8Nq; x1 , x2 ).

(104) 2f 1 (a, b; cNq, x1 5q x2 ) 4˜q (b) F 1 (a; b, b; cNq; x1 , x2 ).

(105) 2f 1 (a, b; cNq, x1 5q x2 ) 4D q (c) F 4 (a, b; c, cNq; x1 , x2 ).

(106) 2f 1 (a, b; cNq, x1 5q x2 ) 4

4˜q (b) D q (c) F 2 (a; b, b; c, cNq; x1 , x2 ).

(107) 2f 1 (a, b; cNq, [x1 1x2 ]q ) 4 a : b; b; 4˜ (b) F 1:1;1 Nq; x , x . q 1:0;1 kc : 2; Q; 1 2l

(108) 2f 1 (a, b; cNq, [x1 1x2 ]q ) 4 a, b: Q, Q; Q; 4D (c) F 2:2;1 Nq; x , x . q 2:1;1 k Q, Q: c; c; 1 2l

(109) 2f 1 (a, b; cNq, [x1 1x2 ]q ) 4 a : b, Q; b; 4˜ (b) D (c) F 1:2;1 Nq; x , x . q q 1:1;1 k Q: c; c; 1 2l For a simple q-hypergeometric series we obtain

1 1 (110) 2 f 1 (a, b; cNq, x1 ) 4 2 f 1 (a, b; c1rNq, x1 ), au 1 1c; qbr ac; qbr and analogous formulae for double series. 228 Thomas Ernst

The following formulas for u 1 and u 2 obtain.

1 (111) F 1 (a; b, b 8; cNq; x1 , x2 ) 4 a2a2u 1 2u 2 11; qbr r (21) aa; qb 2(r)1r(a21) 4 r 2 8 N r r 2 F 1 (a r; b, b ; c q; x1 q , x2 q )q . aa2r; qb2r

(112) a2u 1 ; qbr 2f 1 (a, b; cNq, x1 ) 4 r r (21) aa, b; qb x (r)2r 2 4 r 1 1 1 1 N 2r 2 2f 1 (a r, b r; c r q, x1 q ) q . ac; qbr a; q b, b 8; q 2 2 8 N 4 a b2ra br r r3 (113) a u 1 , u 2 ; qbr F 1 (a; b, b ; c q; x1 , x2) x1 x2 ac; qb2r 2r 2r r 2 2r 3F 1 (a12r; b1r, b 81r; c12rNq; x1 q , x2 q ) q . Combining (111) and (113) we obtain

2u , 2u ; q a 1 2 br 8 N 4 (114) F 1 (a; b, b ; c q; x1 , x2 ) a2a2u 1 2u 2 11; qbr r (21) aa, b, b 8; qb (r)1r(a21) 4 r r r 1 1 81 1 N 2 x1 x2 F 1 (a r; b r, b r; c 2r q; x1 , x2)q . ac; qb2r We obtain the following set of lemmata

Q 2u , 2u ; q 4 a 1 2 br rh r r (115) ˜q (h) ! q e 1 e 2 . 4 r 0 a1, h; qbr Q 2u , 2u ; q 4 a 1 2 br r (116) D q (h) ! q , 4 r 0 a1, 12h2u 1 2u 2 ; qbr

Q r (21) a2u , 2u ; qb ah; qb (r)1rh 4 1 2 r 2r 2 r r (117) D q (h) ! q e 1 e 2 , 4 r 0 a1, h1r21, h1u 1 , h1u 2 ; qbr and

Q 2u , 2u , k2h; q k; q 4 a 1 2 br a b2r rh r r (118) ˜q (h) D q (k) ! q e 1 e 2 , 4 r 0 a1, h, k1r21, k1u 1 , k1u 2 ; qbr Q h2k, 2u , 2u ; q 4 a 1 2 br r (119) ˜q (h) D q (k) ! q , 4 r 0 a1, h, 12k2u 1 2u 2 ; qbr where the last formula is the q-Pfaff-Saalschütz theorem. Some results for q-functions etc. 229

By the same method which was used in [41], we obtain the following q-analogues of the corrected version of Verma [75, (10)-(14)]. The formulas also closely resemble q-analogues of [64, (11), (12), (8), (13), (7)]. When there will be no misunderstanding, we will write G and H for

G1 ; G2 and H1 ; H2 .

THEOREM 5.1.

(a): (g1 ); (g2 ); (120) F A:G1;G2 Nq; x , x 4 B:H1;H2 k 1 2l (b): (h1 ); (h2 );

Q a(g1 ), (g2 ), a; qbr a(a); qb2r 2(r)1ra 1 1 1 4 2 r r A 1:G1 1;G2 1 3 ! q x1 x2 F B11:H111;H211 4 r 0 a1, (h1 ), (h2 ); qbr a(b), a; qb2r

(a12r), Q : (g 1r), a1r; (g 1r), a1r; 3 1 2 Nq; x , x . k 1 1 1r), Q; (h 1r), Q; 1 2l (b 2r), a 2r : (h1 2

Q r (21) a(g ), (g ); qb a(a); qb (r)1ra 4 1 2 r 2r 2 r r 3 (121) ! q x1 x2 4 r 0 a1, (h1 ), (h2 ), a; qbr a(b); qb2r 1 1 1 Q 1 Q A11:G11 (a 2r), a r : (g1 r), ; (g2 r), ; 3F 1 1 Nq; x , x . B 1:H 1 k 1 Q 1 1 1 1 1 2l (b 2r), : (h1 r), a r; (h2 r), a r;

Q a(g ), (g ), g2a, a; qb a(a); qb (r)1ra 4 1 2 r 2r 2 r r 3 (122) ! q x1 x2 4 r 0 a1, (h1 ), (h2 ), g1r21; qbr a(b), a; qb2r 1 1 1 1 1 1 A11:G11 (a 2r), g 2r : (g1 r), a r; (g2 r), a r; 3F 1 1 Nq; x , x . B 1:H 1 k 1 1 1 1 1 1 1 2l (b 2r), a 2r : (h1 r), g 2r; (h2 r), g 2r;

Q r (21) a(g ), (g ); qb a(a); qb 3(r)1ra 4 1 2 r 2r 2 r r 3 (123) ! q x1 x2 4 r 0 a1, (h1 ), (h2 ), a1r21; qbr a(b); qb2r 1 1 1 Q 1 Q A11:G11 (a 2r), a 2r : (g1 r), ; (g2 r), ; 3F 1 1 Nq; x , x . B 1:H 1 k 1 Q 1 1 1 1 1 2l (b 2r), : (h1 r), a 2r; (h2 r), a 2r;

Q r (21) a(g ), (g ), a2g, a; qb a(a); qb (r)1rg 4 1 2 r 2r 2 r r 3 (124) ! q x1 x2 4 r 0 a1, (h1 ), (h2 ), g; qbr a(b), a; qb2r 1 1 1 1 1 1 A11: G11 (a 2r), g r : (g1 r), a r; (g2 r), a r; 3F 1 1 Nq; x , x . B 1:H 1 k 1 1 1 1 1 1 1 2l (b 2r), a 2r : (h1 r), g r; (h2 r), g r; 230 Thomas Ernst

These formulas hold when A1G1GB1H111, A1G2GB1H211, and Nx1N, Nx2N are chosen in such a way that both sides converge [64, p. 49].

PROOF. We will use the following abbreviation. 1 2 1 2 1 1 2 1 2 Uf (21)m(1 H1 G1 B A) n(1 H2 G2 B A) 3 m1n m n 3QE (B2A) 1 (11H 2G ) 1 (11H 2G ) . g g 2 h 1 1 g 2 h 2 2 g2hh First we prove (122). Q 2u , 2u , g2a; q g; q 4 a 1 2 br a b2r ra r r 3 (125) LHS ! q e 1 e 2 4 r 0 a1, a, g1r21, g1u 1 , g1u 2 ; qbr (a), g : (g ), a; (g ), a; 3 A11:G11 1 2 N 4 F B11:H11 k q; x1 , x2l (b), a : (h1 ), g; (h2 ), g; Q 2(r)1ra q 2 g2a; q g; q 4 ! a br a b2r 3 4 r, m, n 0 aa, g1r21, g, g; qbr

A11:G11 (a), g : (g1 ), a; (g2 ), a; 3F 1 1 Nq; x , x 4RHS. B 1:H 1 k 1 1 1 2l (b), a : (h1 ), g r; (h2 ), g r; Equations (120) and (123) follow from (122) by letting gK1Q and aK K1Q respectively. Equation (124) is proved in a similar way. Q 2u , 2u , a2g; q 4 a 1 2 br r A11:G11 3 LHS ! q F B11:H11 4 r 0 a1, a, 12g2u 1 2u 2 ; qbr (a), g : (g ), a; (g ), a; 3 1 2 N 4 y q; x1 , x2z (b), a : (h1 ), g; (h2 ), g; r 1 Q r (2) rg (21) q aa2g; qb ag; qb 1 2 4 ! r m n r U3 4 r, m, n 0 a1, a; qbr (a); q (g ), a; q (g ), a; q x m x n 3 a bm1n a 1 bm a 2 bn 1 2 4 a(b), a; qbm1n a(h1 ), g; qbm a(h2 ), g; qbn a1; qbm2r a1; qbn2r Q (r)1rg (21)r q 2 x r x r aa2g, (g ), a, (g ); qb a(a); qb 4 ! 1 2 1 2 r 2r U3 4 r, m, n 0 a1, (h1 ), g, (h2 ); qbr a(b), a; qb2r m n x x a(a12r), g1r; qb 1 a(g 1r), a1r; qb a(g 1r), a1r; qb 3 1 2 m n 1 m 2 n 4RHS. a(b12r), a12r; qbm1n a(h11r), g1r; qbm a(h21r), g1r; qbn Equation (121) follows from (124) by letting aK1Q. r Some results for q-functions etc. 231

REMARK 11. As pointed out in [75], specialization of parameters in (120) yields the following results from [41]: 33, 36, 39, a corrected version of 42, a corrected version of 43, 45, and 48. Similarly, (121) gives 34, 40, a corrected version of 44 and 46 from the same paper, while (122) gives [2, 6.8 p. 193], and equation 49 from [41]. Moreover, (123) gives [2, p. 194], as well as equations 41 and 47 from [41], and (124) gives 38 and 50 of [41]. Compare [64, p. 51].

6. Conclusion.

There is a typical trend in some of the new results. In the Watson- and Jackson formulas for 8f 7 series, we append an extra tilde in numerator and denominator to get the q-analogue. This is also the case in the LHS triple sum of (92). However, in the RHS double sum of (92) we append two extra tildes in numerator and denominator to get the q-analogue. In (64) and (73) the hypergeometric function argument 4x is repla- ced by x times two tildes in the numerator. The following decomposition of the q-hypergeometric series into even and odd parts is a q-analogue of [68, p. 200-201], and an example of a q- 1 analogue of a hypergeometric function with argument x. 4 THEOREM 6.1. [22, (366), p. 71]

(126) rf s ( (a); (b)Nq, z) 4 A A (a) (a) (a11) (a11) , , , 2 2 2 2 N 2 11s2r 1 44rf 4s13 q, z q A A A y (b) (b) (b11) (b11) 1 1 A z , , , , , ,1 2 2 2 2 2 2 r » (12q aj ) z j41 1(21)11s2r 3 12q s » (12q bj ) j41 A A (a11) (a11) (a12) (a12) , , , 2 2 2 2 2 3(11s2r) 3 f 1 Nq, z q . 4r 4s 3 A A A y (b11) (b11) (b12) (b12) 3 3 A z , , , , , ,1 2 2 2 2 2 2 232 Thomas Ernst

So we have 2 cases where the tilde operator is connected to q-analogues of hypergeometric function arguments expressed as x326n . The extended q-hypergeometric series (28) follows a similar pattern. Hopefully the q-Kampé de Fériet functions defined here will enhance the understanding of this interdisciplinary subject. Further develop- ments along these lines will be given in future papers.

Acknowledgments. I want to thank Per Karlsson who gave some va- lueable comments. I also want to thank Axel Riese for the computer proof. Part of this paper was supported by Kungliga vetenskapsakade- mien.

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Manoscritto pervenuto in redazione l’8 gennaio 2004.