A q-analogue of Kummer s 20 relations

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J. Math. Anal. Appl. 272 (2002 43 54 www.academicpress.com A q-analogue of Kummer s 20 relations Shaun Cooper Institute of Information and Mathematical Sciences, Massey University, Albany Campus, Private Bag 102 904, North Shore Mail Centre, Auckland, New Zealand Received 2 October 2001 Submitted by B.C. Berndt Abstract The 3 φ 2 transformations are used to derive q-analogues of Kummer s 20 relations. 2002 Elsevier Science (USA. All rights reserved. Keywords: Hypergeometric function; Basic hypergeometric series; Euler s transformation; Heine s transformation; Pfaff s transformation; Kummer s 20 relations 1. Introduction The purpose of this article is to give q-analogues of the 20 relations between Kummer s 24 hypergeometric functions. Standard notation for q-series is used throughout see, for example, [1, Chapter 10] or [4, Chapter 1]. An interesting feature of the results is the occurrence of divergent series, namely the 3 φ 1 and 2 φ 0 functions. It is still an open question to assign a meaningful interpretation to these divergent series. They formally reduce to hypergeometric functions when q 1. The paper is organised as follows. In Section 2, the basic transformation formulas for the 2 φ 1 function are given. Three fundamental transformation properties of 3 φ 2 functions are given in Section 3. These formulas are written using Sears [6] q-analogue of Whipple s notation. Only two of these formulas will be used; E-mail address: s.cooper@massey.ac.nz. 0022-247X/02/$ see front matter 2002 Elsevier Science (USA. All rights reserved. PII: S0022-247X(0200130-0

44 S. Cooper / J. Math. Anal. Appl. 272 (2002 43 54 the third is mentioned for completeness. The 20 q-analogues of the relations between Kummer s 24 solutions the main results of this paper are given in Section 4. A proof of one of the 20 results is given in detail in Section 5 and the remaining proofs, which are similar, are summarized in Table 2. The key idea in the proofs is to use limiting properties of 3 φ 2 functions, for example: lim f =abcx, c 0 3 φ 2 ( a,b,c e,f ; ef abc We conclude with some remarks in Section 6. = 2 φ 1 ( a,b. 2. Basic properties of the 2 φ 1 function The following formulas will be used throughout: = (e/a, e/b ( a,b e ; e ab ( a,b, (1 (e, e/ab = (b, ax ( e/b,x (e, x ax ; b (2 = (e/b, bx (,b ; e/b (3 (e, x bx = ( ( e/a,e/b ; (4 (x e = (ax ( a,e/b 2φ 2 (x e,ax ; bx (5 = (ax, bx ( x, 2φ 2 (e, x ax,bx ; e (6 ( a,e/b,0 e,eq/bx. (7 = ( (bx Equation (1 is Heine s q-analogue of Gauss theorem ([1, p. 522], [2, p. 68], [4, p. 10]. Equations (2, (3, and (4 are Heine s transformation, the iterate of Heine s transformation, and the q-analogue of Euler s transformation, respectively ([1, pp. 521 524], [4, pp. 9 10]. Equation (5 is Jackson s q-analogue of Pfaff s transformation ([1, (10.10.12], [4, (1.5.4], and Eq. (6 appears in [6] as the function Y(1, 6 in Table IIA. Equation (7 is due to Jackson [4, p. 241, (III.5]. 3. Transformation properties of 3 φ 2 functions Let r 0, r 1, r 2, r 3, r 4, r 5 be six parameters such that r 0 r 1 r 2 r 3 r 4 r 5 = 1. With these parameters, associate numbers α lmn and β mn such that

S. Cooper / J. Math. Anal. Appl. 272 (2002 43 54 45 Table 1 α 012 = q/c α 123 = ef /abc β 01 = q 2 bcf β 20 = ef /ac β 40 = e α 013 = q/b α 124 = e/c β 02 = q 2 acf β 21 = qb/a β 41 = qbc/f α 014 = qa/f α 125 = f/c β 03 = q 2 abf β 23 = qb/c β 42 = qac/f α 015 = qa α 134 = e/b β 04 = q 2 β 24 = qf/ac β 43 = qab/f α 023 = q/a α 135 = f/b β 05 = q 2 /f β 25 = qe/ac β 45 = qe/f α 024 = qb/f α 145 = a β 10 = ef /bc β 30 = ef /ab β 50 = f α 025 = qb α 234 = e/a β 12 = qa/b β 31 = qc/a β 51 = qbc α 034 = qc/f α 235 = f/a β 13 = qa/c β 32 = qc/b β 52 = qac α 035 = qc α 245 = b β 14 = qf/bc β 34 = qf/ab β 53 = qab α 045 = qabcf α 345 = c β 15 = qe/bc β 35 = qe/ab β 54 = qf α lmn = q 1/2 r l r m r n, β mn = qr m /r n. Let (i; j,k; l,m,n be any permutation of (0, 1, 2, 3, 4, 5 and let ( αjkl,α jkm,α jkn F(i; j,k= (β ji,β ki,α lmn 3 φ 2 ; α lmn β ji,β ki. (8 Clearly F(i; j,k= F(i; k,j. (9 Set α 145 = a, α 245 = b, α 345 = c, β 40 = e, β 50 = f. Equivalently, r0 3 = q5/2 abc e 2 f 2, r1 3 = aef q 1/2 b 2 c 2, r3 2 = bef q 1/2 a 2 c 2, r3 3 = cef q 1/2 a 2 b 2, r3 4 = abce q 1/2 f 2, r3 5 = abcf q 1/2 e 2. All of the α s and β s can be expressed in terms of a, b, c, e, f and q (cf. Table 1. With this notation, ( a,b,c F(0; 4, 5 = (e,f,ef/abc 3 φ 2 e,f ; ef. (10 abc 3.1. Two-term transformations Hall s ([1, (10.10.8], [4, (3.2.10], [5], [6, p. 173, statement I] formula ( a,b,c (e,f,ef/abc 3 φ 2 e,f ; ef abc ( e/a,f/a,ef/abc = (a,ef/ab,ef/ac 3 φ 2 ; a ef/ab, ef/ac becomes in this notation simply F(0; 4, 5 = F(0; 2, 3. (11

46 S. Cooper / J. Math. Anal. Appl. 272 (2002 43 54 Interchanging r 4 and r 1,wefindthat F(0; 1, 5 = F(0; 2, 3 and thus F(0; 1, 5 = F(0; 4, 5. (12 By repeatedly interchanging r u and r v, u, v {1, 2, 3, 4, 5}, and using the symmetry (9, we find that all 20 expressions F(0; j,k are equal, and therefore may be denoted by F(0. In Eq. (11, interchanging r 0 and r 1 we find that F(1; 4, 5 = F(1; 2, 3. Proceeding as before and interchanging the r s and using symmetry, we find that all 20 expressions F(1; j,k are equal, and may be denoted by F(1. Similarly, for each of the remaining values i = 2, 3, 4, 5, the 20 functions F(i; j,kare equal and may be denoted by F(i. In summary, the 120 functions F(i; j,k may be divided into 6 classes, each consisting of 20 identical functions. Specifically, F(i= F(i; j,k for any permutation (i; j,k; l,m,n of (0, 1, 2, 3, 4, 5. 3.2. Three-term relations The three-term relation ([4, (3.3.1], [6, p. 173 II(a] in this notation is equivalent to F(i= (α ( klm,α kln,α kmn,β ji αjlm,α jln,α jmn (β kj /q + (α jlm,α jln,α jmn,β ki (β jk /q β ji,β jk ( αklm,α kln,α kmn β ki,β kj, (13 where (i,j,k,l,m,n is any permutation of (0, 1, 2, 3, 4, 5. Observe that the right-hand side is symmetric in j and k, andalsoinl, m and n. Consequently, there are 6 ( 5 2 = 60 equations of this type. Although we shall not need it, the three-term relation ([4, (3.3.3], [6, p. 173 III(b], written in this notation, is 0 = (r k r j (β kj,β jk,α ilm,α iln,α imn F(i + (r i r k (β ik,β ki,α jlm,α jln,α jmn F(j + (r j r i (β ji,β ij,α klm,α kln,α kmn F(k. (14 Any permutation of the indices 0,...,5 is legitimate, and so there are ( 6 3 = 20 equations of this type relating any three of the functions F(0,...,F(5.

S. Cooper / J. Math. Anal. Appl. 272 (2002 43 54 47 Equations (11 and (12 are q-analogues of [2, Section 3.6, (1 and (2], while (13 and (14 are q-analogues of [2, Section 3.7, (2 and (3], respectively. 4. The 20 relations In this section, q-analogues of the 20 relations between Kummer s solutions are given. For the purpose of ease of reference, the same equation numbering system as in [3, pp. 106 108] is used; i.e., our Eq. (25 is the q-analogue of Eq. (25 on p. 106 of [3], etc. ( a,b,q/x qab/ e = (qa,e ( (x, q/x 1/b, bx ; e ( a,b (qab, e/b (bx, q/bx x + (b, qa/b ( (, qe/ q,x/q (qab, e/b (bx/q, q 2 /bx ; e x, (25 ( a,qa qa/b e ( a,b,q/x qab/ e = (qb,e ( (x, q/x 1/a, ax ; e ( a,b (qab, e/a (ax, q/ax x + (a, qb/a ( (, qe/ q,x/q (qab, e/a (ax/q, q 2 /ax ; e x ( b,qb qb/a e ( (x ( e/a,e/b,eq/ eq/ab, (26 ; e 2 = (q/a, e (, qe/ (qe/ab, b (ax, q/ax ( a,b + (qb/a,e/b (, qe/ (qe/ab, b (ax/q, q 2 /ax ( b,bq bq/a e ( b,ax ; e2 ( q,q ; e2, (27

48 S. Cooper / J. Math. Anal. Appl. 272 (2002 43 54 ( qa,qb,q/x ; ex qab q 2 = (a, ( q2 (x, q/x e/qb,bx/q (qab, q/b (qbx, e/bx ( qa,qb q 2 ; x + (qb, qa/b (, qe/ (qab, q/b (bx,qe/bx ( (x ( qa,a qa/b e ( e/a,e/b,eq/ eq/ab ex ( e/q,x/q ex, (28 ; e 2 = (q/b, e (, qe/ (qe/ab, a (bx, q/bx ( a,b + (qa/b,e/a (, qe/ (qe/ab, a (bx/q, q 2 /bx ( a,aq aq/b e ( qa,qb,q/x qab ; ex q 2 ( a,bx ; e2 ( q,q ; e2, (29 = (b, ( q2 (x, q/x e/qa,ax/q (qab, q/a (qax, e/ax ( qa,qb q 2 ; x + (qa, qb/a (, qe/ (qab, q/a (ax,qe/ax ( qb,b qb/a e ( q/a,q/b,qe/ qe/ab ; q 2 ex ( e/q,x/q ex, (30 = (e/a, ( q2 (x, qe/ b/q,ax/q (qe/ab,qb (qax, e/ax

S. Cooper / J. Math. Anal. Appl. 272 (2002 43 54 49 ( qa,qb q 2 ; x + (q/b, qb/a ( (x, qe/ e/q,q (qe/ab,qb (ax,qe/ax ( qb,b qb/a e ( q/a,q/b,qe/ qe/ab ; q 2, (31 = (e/b, ( q2 (x, qe/ a/q,bx/q (qe/ab,qa (qbx, e/bx ( qa,qb q 2 ; x + (q/a, qa/b ( (x, qe/ e/q,q (qe/ab,qa (bx,qe/bx ( a,b ( a,qa qa/b e = (e/a, e/b (e, e/ab + (a, b (e, ab ( (x ( a,b = (e/a, b (e, b/a (ax, q/ax (x, q/x, (32 ( a,b, 0,qab ( e/a,e/b,x 0,eq/ab + (a, e/b (bx, q/bx (e, a/b (x, q/x ( a,b,q/x qab/ e = (qa, qb (q, qab + (a, b (qab, e/q ( a,qa qa/b e ( b,qb qb/a e ( a,b ( e/q,q/x ; x e, (33, (34 ( qa,qb q 2 ; x, (35

50 S. Cooper / J. Math. Anal. Appl. 272 (2002 43 54 ( a,b,q/x qab/ e = (qb,b (b/a, qab + (a, qa (a/b, qab ( 1/a, ax ; e ( a,qa x qa/b e ( 1/b, bx ; e ( b,qb x qb/a e ( a,qa qa/b e = (q/b, qa ( (bx,qe/bx a,b (q, qa/b (, qe/ + (a, e/b (bx/q, q 2 ( /bx qa,qb (e/q, qa/b (, qe/ q 2 ( a,qa qa/b e = (a, qa ( (q/x q/b,e/b,qe/ (ab, qa/b (qe/ 0,qe/ab + (q/b, e/b ( a, qa, q/x (qa/b,e/ab qab,0 ( b,qb qb/a e ( a,b, (36 ; x, (37, (38 = (q/a, qb (ax,qe/ax (q, qb/a (, qe/ + (b, e/a (ax/q, q 2 ( /ax qa,qb (e/q, qb/a (, qe/ q 2 ( b,qb qb/a e = (q/a, e/a ( b, qb, q/x (qb/a,e/ab qab,0 + (b, qb ( (q/x q/a,e/a,qe/ (ab, qb/a (qe/ 0,qe/ab ( qa,qb q 2 ; x = (q/a, q/b (q 2, e/ab ( aq,bq, 0,qab ; x, (39, (40

S. Cooper / J. Math. Anal. Appl. 272 (2002 43 54 51 + (qa, qb ( ( q/a,q/b,x (q 2, ab (x 0,eq/ab, (41 ( qa,qb q 2 ; x = (q/a, qb (q 2, b/a (qax, e/ax (x, q/x + (q/b, qa (q 2, a/b (qbx, e/bx (x, q/x ( ( e/a,e/b,eq/ ; (x eq/ab e 2 = (q/a, q/b ( a,b (q, qe/ab + (e/a, e/b ( e/q,eq/ (qe/ab, e/q ( a,qa qa/b e ( b,qb qb/a e ; e 2, (42 ( aq,bq q 2 ; x, (43 ( (q/x e/a,e/b,eq/ ; (eq/ eq/ab e 2 = (e/a, q/a ( ( b,ax ; e2 a,qa (qe/ab, b/a qa/b e + (q/b, e/b ( a,bx ; e2 (a/b, qe/ab ( b,qb qb/a ; eq. (44 5. Proofs 5.1. Proof of (25 Take (i; j,k; l,m,n = (3; 0, 2; 1, 4, 5 in (13 and use F(3 = F(3; 4, 5 to get ( α045,α 145,α 245 (β 43,β 53,α 012 3 φ 2 ; α 012 β 43,β 53 = (α ( 124,α 125,α 245,β 03 α014,α 015,α 045 (β 20 /q β 03,β 02

52 S. Cooper / J. Math. Anal. Appl. 272 (2002 43 54 + (α ( 014,α 015,α 045,β 23 α124,α 125,α 245. (β 02 /q β 23,β 20 In terms of the parameters a,b,c,e,f and q,thisis ( qabcf,a,b qab/f,qab c = (e/c,f/c,b,q 2 abf (q/c,qab/f,qab,ef/qac + (qa/f,qa,qabcf,qb/c (q/c,qab/f,qab,qacf Set f = abcx to get ( q/x,a,b qe/cx,qab c = (b, (e/c, q 2 /cx (qab, bx/q (q/c,qe/cx ( qe/bcx,qa,q/x 3 φ 2 q 2 /cx, q 2 /bx + (qa, q/x (qb/c,qe/bcx (qab, q/bx (q/c,qe/cx ( e/c,,b 3 φ 2 qb/c,bx ( qa/f,qa,qabcf q 2 abf,q 2 acf ( e/c,f/c,b qb/c,ef/ac.. (45 Next, observe that as c 0, ( q/x,a,b qe/cx,qab ( q/x,a,b 3 φ 1 c qab/, (46 e ( ( qe/bcx,qa,q/x qa, q/x q 2 /cx, q 2 2 φ 1 /bx q 2 /bx ; e b = (qe/, qa/b ( a,qa (q 2 /bx, e/b qa/b e, (47 ( ( e/c,,b,b 2 φ 1 ; e qb/c,bx bx b = (x, e ( a,b (bx, e/b. (48 The iterate of Heine s transformation (3 has been used to transform the 2 φ 1 functions in (47 and (48. Furthermore, using (1 we have (e/c, q 2 ( /cx q,x/q = 2 φ 1 e ( q,x/q 2 φ 0 (q/c,qe/cx q/c cx ; e (49 x

S. Cooper / J. Math. Anal. Appl. 272 (2002 43 54 53 Table 2 Eq. (i; j,k; l,m,n Alternative proof by change of variable (25 (3; 0, 2; 1, 4, 5 (26 (3; 0, 1; 2, 4, 5 a b in (25 (27 (5; 0, 1; 2, 3, 4 a e/a, b e/b, x in (25 (28 (3; 2, 4; 0, 1, 5 a qa, b qb, e q 2 in (25 (29 (5; 0, 2; 1, 3, 4 a e/b, b e/a, x in (25 (30 (3; 1, 4; 0, 2, 5 a qb, b qa, e q 2 in (25 (31 (5; 1, 4; 0, 2, 3 a q/a, b q/b, e q 2, x in (25 (32 (5; 2, 4; 0, 1, 3 a q/b, b q/a, e q 2, x in (25 (33 (0; 3, 5; 1, 2, 4 (34 (0; 1, 2; 3, 4, 5 (35 (3; 0, 4; 1, 2, 5 (36 (3; 1, 2; 0, 4, 5 (37 (2; 0, 4; 1, 3, 5 (38 (2; 3, 5; 0, 1, 4 (39 (1; 0, 4; 2, 3, 5 a b in (37 (40 (1; 3, 5; 0, 2, 4 a b in (38 (41 (4; 3, 5; 0, 1, 2 a qa, b qb, e q 2 in (33 (42 (4; 1, 2; 0, 3, 5 a qa, b qb, e q 2 in (34 (43 (5; 0, 4; 1, 2, 3 a e/a, b e/b, x in (35 (44 (5; 1, 2; 0, 3, 4 a e/a, b e/b, x in (36 and similarly (qb/c,qe/bcx (q/c,qe/cx = 2 φ 1 ( 1/b, bx q/c e cx ( 1/b, bx 2 φ 0 ; e. x (50 Equation (25 now follows by taking the limit as c 0 in (45 and using Eqs. (46 (50. 5.2. Remaining proofs The remaining equations can be proved in the same way. Equations (26 (32 can also be obtained from (25 just by change of variable. A summary of the proofs is presented in Table 2. 6. Remarks Equation (33 is due to Sears [6, p. 178, II(a]. It also can be obtained by taking θ r = (p r in [6, Theorem 4] and then replacing all occurrences of p with q. Equation (34 is due to Watson [7, p. 285, Eq. (7]. It also appears in [6, p. 178, III(d] and [4, p. 106, Eq. (4.3.2]. Equivalent forms of (34 are given

54 S. Cooper / J. Math. Anal. Appl. 272 (2002 43 54 in [6, p. 178, II(c] and [4, p. 92, Ex. 3.8]; the equivalence follows immediately using the transformation formula (7. Equation (39 is due to Sears [6, p. 178, III(c]. An equivalent form (use Heine s transformation (2 is given in [4, p. 64, Eq. (3.3.5]. In Eq. (39, first interchange a and b, then replace x with x/b and let b to get (qa (q ( (x, qe/x a 1φ 1 (ax,qe/ax + (a (e/q ( a,qa = 2 φ 1 e 0 ax (x/q, q 2 ( /x qa 1φ 1 (ax,qe/ax q 2 x e. (51 This formula is a q-analogue of [1, p. 192, Eq. (4.1.13]. Note, however, that the right-hand side of Eq. (4.1.13 in [1] is an asymptotic expansion which does not converge, while the q-analogue (51 is an equality among convergent series. References [1] G. Andrews, R. Askey, R. Roy, Special Functions, Cambridge University Press, 1999. [2] W.N. Bailey, Generalized Hypergeometric Series, Cambridge University Press, 1935. [3] A. Erdelyi (Ed., Higher Transcendental Functions, Vol. 1, McGraw Hill, 1953; reprinted Robert E. Krieger Publishing Company, 1981. [4] G. Gasper, M. Rahman, Basic Hypergeometric Series, Cambridge University Press, 1990. [5] N.A. Hall, An algebraic identity, J. London Math. Soc. 11 (1936 276. [6] D.B. Sears, On the transformation theory of basic hypergeometric functions, Proc. London Math. Soc. (2 53 (1951 158 180. [7] G.N. Watson, The continuations of functions defined by generalised hypergeometric series, Trans. Cambridge Philos. Soc. XXI (1910 281 299.

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