A q-analogue of Kummer s 20 relations

Transcription

1 J. Math. Anal. Appl. 272 ( A q-analogue of Kummer s 20 relations Shaun Cooper Institute of Information and Mathematical Sciences, Massey University, Albany Campus, Private Bag , 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 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; address: s.cooper@massey.ac.nz X/02/$ see front matter 2002 Elsevier Science (USA. All rights reserved. PII: S X(

2 44 S. Cooper / J. Math. Anal. Appl. 272 ( 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 ], [4, pp. 9 10]. Equation (5 is Jackson s q-analogue of Pfaff s transformation ([1, ( ], [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

3 S. Cooper / J. Math. Anal. Appl. 272 ( 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, ( ], [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

4 46 S. Cooper / J. Math. Anal. Appl. 272 ( 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, 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.

5 S. Cooper / J. Math. Anal. Appl. 272 ( 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 ] 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

6 48 S. Cooper / J. Math. Anal. Appl. 272 ( ( 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

7 S. Cooper / J. Math. Anal. Appl. 272 ( ( 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

8 50 S. Cooper / J. Math. Anal. Appl. 272 ( ( 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

9 S. Cooper / J. Math. Anal. Appl. 272 ( (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 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

10 52 S. Cooper / J. Math. Anal. Appl. 272 ( (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

11 S. Cooper / J. Math. Anal. Appl. 272 ( 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 ( 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 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

12 54 S. Cooper / J. Math. Anal. Appl. 272 ( 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. ( 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, [2] W.N. Bailey, Generalized Hypergeometric Series, Cambridge University Press, [3] A. Erdelyi (Ed., Higher Transcendental Functions, Vol. 1, McGraw Hill, 1953; reprinted Robert E. Krieger Publishing Company, [4] G. Gasper, M. Rahman, Basic Hypergeometric Series, Cambridge University Press, [5] N.A. Hall, An algebraic identity, J. London Math. Soc. 11 ( [6] D.B. Sears, On the transformation theory of basic hypergeometric functions, Proc. London Math. Soc. (2 53 ( [7] G.N. Watson, The continuations of functions defined by generalised hypergeometric series, Trans. Cambridge Philos. Soc. XXI (