The 3φ2 transformations are used to derive q-analogues ofthe relations amongst Kummer’s 24
solutions

Cooper, Shaun

English

Massey Research Online

Res. Lett. Inf. Math. Sci., (2001) 2, 1–10Available online at http://www.massey.ac.nz/∼wwims/rlims/On the q-analogue of Kummer’s 24 solutionsShaun CooperI.I.M.S., Massey University Albany Campus, Auckland, New Zealands.cooper@massey.ac.nzAbstractThe 3φ2 transformations are used to derive q-analogues of the relations amongst Kummer’s 24solutions.1 IntroductionThe purpose of this article is to give q-analogues of the 20 relations between Kummer’s 24 hyper-geometric functions.Standard notation for q-series is used throughout - see, for example, [1, Chapter 10] or [4, Chapter1].An interesting feature of the results is the occurrence of divergent series, namely the 3φ1 and2φ0 functions. It is still an open question to assign a meaningful interpretation to these divergentseries. They formally reduce to hypergeometric functions when q → 1.The paper is organised as follows. In section 2, the basic transformation formulas for the 2φ1function are given. Three fundamental transformation properties of 3φ2 functions are given insection 3. These formulas are written using Sears’ [6] q-analogue of Whipple’s notation. Only twoof these formulas will be used; the third is mentioned for completeness. The 20 q-analogues of therelations 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 aresimilar, are summarized in a table. The key idea in the proofs is to use limiting properties of 3φ2functions, for example:limf=abcx/e, c→0 3φ2(a, b, ce, f;efabc)= 2φ1(a, be;x).2 Basic properties of the 2φ1 functionThe following formulas will be used throughout.2φ1(a, be;eab)=(e/a, e/b; q)∞(e, e/ab; q)∞; (1)2φ1(a, be;x)=(b, ax; q)∞(e, x; q)∞2φ1(e/b, xax; b)(2)=(e/b, bx; q)∞(e, x; q)∞2φ1(abx/e, bbx; e/b)(3)=(abx/e; q)∞(x; q)∞2φ1(e/a, e/be; abx/e)(4)=(ax; q)∞(x; q)∞2φ2(a, e/be, ax; bx)(5)2 R.L.I.M.S. Vol. 2, May 2001=(ax, bx; q)∞(e, x; q)∞2φ2(x, abx/eax, bx; e)(6)=(abx/e; q)∞(bx/e; q)∞3φ2(a, e/b, 0e, eq/bx; q). (7)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-analogueof Euler’s transformation, respectively [1, pp. 521–524], [4, pp. 9–10]. Equation (5) is Jackson’sq-analogue of Pfaﬀ’s transformation [1, (10.10.12)], [4, (1.5.4)], and equation (6) appears in [6] asthe function Y (1, 6) in Table IIA. Equation (7) is due to Jackson [4, p. 241, (III.5)].3 Transformation properties of 3φ2 functionsLet r0, r1, r2, r3, r4, r5 be six parameters such that r0r1r2r3r4r5 = 1. With these parameters,associate numbers αlmn and βmn such thatαlmn = q1/2rlrmrnβmn = qrm/rn.Let (i; j, k; l, m, n) be any permutation of (0, 1, 2, 3, 4, 5). The semicolons indicate that permutationswith the second and third entries interchanged are regarded as equivalent, as are permutations withthe last three entries in any order. For example, (2; 1, 4; 5, 0, 3) is equivalent to (2; 4, 1; 5, 0, 3), andboth are equivalent to (2; 1, 4; 0, 3, 5). Consequently, it is only necessary to specify the ﬁrst threeentries, in which case (i; j, k) can be used to represent (i; j, k; l, m, n). Accordingly, letF (i; j, k) = (βji, βki, αlmn; q)∞3φ2(αjkl, αjkm, αjknβji, βki;αlmn). (8)Set α145 = a, α245 = b, α345 = c, β40 = e, β50 = f . Equivalently,r30 =q5/2abce2f2r31 =aefq1/2b2c2r32 =befq1/2a2c2r33 =cefq1/2a2b2r34 =abceq1/2f2r35 =abcfq1/2e2.All of the α’s and β’s can be expressed in terms of a, b, c, d, e and q:S. Cooper, On the q-analogue of Kummer’s 24 solutions 3α012 = q/c α123 = ef/abc β01 = q2bc/ef β20 = ef/ac β40 = eα013 = q/b α124 = e/c β02 = q2ac/ef β21 = qb/a β41 = qbc/fα014 = qa/f α125 = f/c β03 = q2ab/ef β23 = qb/c β42 = qac/fα015 = qa/e α134 = e/b β04 = q2/e β24 = qf/ac β43 = qab/fα023 = q/a α135 = f/b β05 = q2/f β25 = qe/ac β45 = qe/fα024 = qb/f α145 = a β10 = ef/bc β30 = ef/ab β50 = fα025 = qb/e α234 = e/a β12 = qa/b β31 = qc/a β51 = qbc/eα034 = qc/f α235 = f/a β13 = qa/c β32 = qc/b β52 = qac/eα035 = qc/e α245 = b β14 = qf/bc β34 = qf/ab β53 = qab/eα045 = qabc/ef α345 = c β15 = qe/bc β35 = qe/ab β54 = qf/eWith this notation,F (0; 4, 5) = (e, f, ef/abc; q)∞3φ2(a, b, ce, f;efabc). (9)Two term transformationsHall’s [1, (10.10.8)], [4, (4.2.10)], [5], [6, (p. 173, statement I)] formula(e, f, ef/abc; q)∞3φ2(a, b, ce, f;efabc)= (a, ef/ab, ef/ac; q)∞3φ2(e/a, f/a, ef/abcef/ab, ef/ac; a)becomes in this notation simplyF (0; 4, 5) = F (0; 2, 3). (10)Interchanging r4 and r1, we ﬁnd thatF (0; 1, 5) = F (0; 2, 3)and thusF (0; 1, 5) = F (0; 4, 5). (11)Accordingly all of the permutations of the indices 1 to 5 are legitimate, and we see that all tenexpressions F (0; v, w) are equal, and may be denoted by F (0). For every value u = 1, · · · , 5, theten functions F (u; v, w) are equal and may be denoted by F (u).Three term relationsThe three term relation [4, (3.3.1)], [6, (p. 173 II(a))] in this notation is equivalent toF (i) =(αklm, αkln, αkmn, βji; q)∞(βkj/q; q)∞3φ2(αjlm, αjln, αjmnβji, βjk; q)+(αjlm, αjln, αjmn, βki; q)∞(βjk/q; q)∞3φ2(αklm, αkln, αkmnβki, βkj; q), (12)where (i, j, k, l, m, n) is any permutation of (0, 1, 2, 3, 4, 5). Observe that the right hand side is sym-metric in j and k, and also in l, m and n. Consequently, there are 6×(52) = 60 equations of this type.Although we shall not need it, the three term relation [4, (3.3.3)], [6, (p. 173 III(b))], writtenin this notation, is0 = (rk − rj)(βkj , βjk, αilm, αiln, αimn; q)∞F (i)+ (ri − rk)(βik, βki, αjlm, αjln, αjmn; q)∞F (j)+ (rj − ri)(βji, βij , αklm, αkln, αkmn; q)∞F (k). (13)4 R.L.I.M.S. Vol. 2, May 2001Any permutation of the indices 0, · · · , 5 is legitimate, and so there are (63) = 20 equations of thistype relating any three of the functions F (0), · · · , F (5).Equations (10) and (11) are q-analogues of [2, section 3.6, (1) and (2)], while (12) and (13) areq-analogues of [2, section 3.7, (2) and (3)], respectively.4 The 20 relationsIn this section, q-analogues of the 20 relations between Kummer’s solutions are given. For thepurpose of ease of reference, the same equation numbering system as [3, pp. 106 – 108] is used,i.e., our equation (25) is the q-analogue of equation (25) on page 106 of [3], etc.3φ1(a, b, q/xqab/e;xe)=(qa/e, e)∞(qab/e, e/b)∞(x, q/x)∞(bx, q/bx)∞2φ0(1/b, bx/e− ;ex)2φ1(a, be;x)+(b, qa/b)∞(qab/e, e/b)∞(abx/e, qe/abx)∞(bx/q, q2/bx)∞2φ0(q/e, x/q− ;ex)2φ1(a, qa/eqa/b;qeabx)(25)3φ1(a, b, q/xqab/e;xe)=(qb/e, e)∞(qab/e, e/a)∞(x, q/x)∞(ax, q/ax)∞2φ0(1/a, ax/e− ;ex)2φ1(a, be;x)+(a, qb/a)∞(qab/e, e/a)∞(abx/e, qe/abx)∞(ax/q, q2/ax)∞2φ0(q/e, x/q− ;ex)2φ1(b, qb/eqb/a;qeabx)(26)(abx/e)∞(x)∞3φ1(e/a, e/b, eq/abxeq/ab;abxe2)=(q/a, e)∞(qe/ab, b)∞(abx/e, qe/abx)∞(ax, q/ax)∞2φ0(b/e, ax/e− ;e2abx)2φ1(a, be;x)+(qb/a, e/b)∞(qe/ab, b)∞(abx/e, qe/abx)∞(ax/q, q2/ax)∞2φ0(q/e, abx/eq− ;e2abx)2φ1(b, bq/ebq/a;qeabx)(27)3φ1(qa/e, qb/e, q/xqab/e;exq2)=(a, q2/e)∞(qab/e, q/b)∞(x, q/x)∞(qbx/e, e/bx)∞2φ0(e/qb, bx/q− ;q2ex)2φ1(qa/e, qb/eq2/e;x)+(qb/e, qa/b)∞(qab/e, q/b)∞(abx/e, qe/abx)∞(bx/e, qe/bx)∞2φ0(e/q, x/q− ;q2ex)2φ1(qa/e, aqa/b;qeabx)(28)(abx/e)∞(x)∞3φ1(e/a, e/b, eq/abxeq/ab;abxe2)=(q/b, e)∞(qe/ab, a)∞(abx/e, qe/abx)∞(bx, q/bx)∞2φ0(a/e, bx/e− ;e2abx)2φ1(a, be;x)+(qa/b, e/a)∞(qe/ab, a)∞(abx/e, qe/abx)∞(bx/q, q2/bx)∞2φ0(q/e, abx/eq− ;e2abx)2φ1(a, aq/eaq/b;qeabx)S. Cooper, On the q-analogue of Kummer’s 24 solutions 5(29)3φ1(qa/e, qb/e, q/xqab/e;exq2)=(b, q2/e)∞(qab/e, q/a)∞(x, q/x)∞(qax/e, e/ax)∞2φ0(e/qa, ax/q− ;q2ex)2φ1(qa/e, qb/eq2/e;x)+(qa/e, qb/a)∞(qab/e, q/a)∞(abx/e, qe/abx)∞(ax/e, qe/ax)∞2φ0(e/q, x/q− ;q2ex)2φ1(qb/e, bqb/a;qeabx)(30)3φ1(q/a, q/b, qe/abxqe/ab;abxq2)=(e/a, q2/e)∞(qe/ab, qb/e)∞(x, qe/abx)∞(qax/e, e/ax)∞2φ0(b/q, ax/q− ;q2abx)2φ1(qa/e, qb/eq2/e;x)+(q/b, qb/a)∞(qe/ab, qb/e)∞(x, qe/abx)∞(ax/e, qe/ax)∞2φ0(e/q, abx/eq− ;q2abx)2φ1(qb/e, bqb/a;qeabx)(31)3φ1(q/a, q/b, qe/abxqe/ab;abxq2)=(e/b, q2/e)∞(qe/ab, qa/e)∞(x, qe/abx)∞(qbx/e, e/bx)∞2φ0(a/q, bx/q− ;q2abx)2φ1(qa/e, qb/eq2/e;x)+(q/a, qa/b)∞(qe/ab, qa/e)∞(x, qe/abx)∞(bx/e, qe/bx)∞2φ0(e/q, abx/eq− ;q2abx)2φ1(a, qa/eqa/b;qeabx)(32)2φ1(a, be;x)=(e/a, e/b)∞(e, e/ab)∞3φ2(a, b, abx/e0, qab/e; q)+(a, b)∞(e, ab/e)∞(abx/e)∞(x)∞3φ2(e/a, e/b, x0, eq/ab; q)(33)2φ1(a, be;x)=(e/a, b)∞(e, b/a)∞(ax, q/ax)∞(x, q/x)∞2φ1(a, qa/eqa/b;qeabx)+(a, e/b)∞(e, a/b)∞(bx, q/bx)∞(x, q/x)∞2φ1(b, qb/eqb/a;qeabx)(34)3φ1(a, b, q/xqab/e;xe)=(qa/e, qb/e)∞(q/e, qab/e)∞2φ1(a, be;x)+(a, b)∞(qab/e, e/q)∞2φ0(e/q, q/x− ;xe)2φ1(qa/e, qb/eq2/e;x)(35)6 R.L.I.M.S. Vol. 2, May 20013φ1(a, b, q/xqab/e;xe)=(qb/e, b)∞(b/a, qab/e)∞2φ0(1/a, ax/e− ;ex)2φ1(a, qa/eqa/b;qeabx)+(a, qa/e)∞(a/b, qab/e)∞2φ0(1/b, bx/e− ;ex)2φ1(b, qb/eqb/a;qeabx)(36)2φ1(a, qa/eqa/b;qeabx)=(q/b, qa/e)∞(q/e, qa/b)∞(bx/e, qe/bx)∞(abx/e, qe/abx)∞2φ1(a, be;x)+(a, e/b)∞(e/q, qa/b)∞(bx/q, q2/bx)∞(abx/e, qe/abx)∞2φ1(qa/e, qb/eq2/e;x)(37)2φ1(a, qa/eqa/b;qeabx)=(a, qa/e)∞(ab/e, qa/b)∞(q/x)∞(qe/abx)∞3φ2(q/b, e/b, qe/abx0, qe/ab; q)+(q/b, e/b)∞(qa/b, e/ab)∞3φ2(a, qa/e, q/xqab/e, 0; q)(38)2φ1(b, qb/eqb/a;qeabx)=(q/a, qb/e)∞(q/e, qb/a)∞(ax/e, qe/ax)∞(abx/e, qe/abx)∞2φ1(a, be;x)+(b, e/a)∞(e/q, qb/a)∞(ax/q, q2/ax)∞(abx/e, qe/abx)∞2φ1(qa/e, qb/eq2/e;x)(39)2φ1(b, qb/eqb/a;qeabx)=(q/a, e/a)∞(qb/a, e/ab)∞3φ2(b, qb/e, q/xqab/e, 0; q)+(b, qb/e)∞(ab/e, qb/a)∞(q/x)∞(qe/abx)∞3φ2(q/a, e/a, qe/abx0, qe/ab; q)(40)2φ1(qa/e, qb/eq2/e;x)=(q/a, q/b)∞(q2/e, e/ab)∞3φ2(aq/e, bq/e, abx/e0, qab/e; q)+(qa/e, qb/e)∞(q2/e, ab/e)∞(abx/e)∞(x)∞3φ2(q/a, q/b, x0, eq/ab; q)(41)2φ1(qa/e, qb/eq2/e;x)S. Cooper, On the q-analogue of Kummer’s 24 solutions 7=(q/a, qb/e)∞(q2/e, b/a)∞(qax/e, e/ax)∞(x, q/x)∞2φ1(a, qa/eqa/b;qeabx)+(q/b, qa/e)∞(q2/e, a/b)∞(qbx/e, e/bx)∞(x, q/x)∞2φ1(b, qb/eqb/a;qeabx)(42)(abx/e)∞(x)∞3φ1(e/a, e/b, eq/abxeq/ab;abxe2)=(q/a, q/b)∞(q/e, qe/ab)∞2φ1(a, be;x)+(e/a, e/b)∞(qe/ab, e/q)∞2φ0(e/q, eq/abx− ;abxe2)2φ1(aq/e, bq/eq2/e;x)(43)(q/x)∞(eq/abx)∞3φ1(e/a, e/b, eq/abxeq/ab;abxe2)=(e/a, q/a)∞(qe/ab, b/a)∞2φ0(b/e, ax/e− ;e2abx)2φ1(a, qa/eqa/b;qeabx)+(q/b, e/b)∞(a/b, qe/ab)∞2φ0(a/e, bx/e− ;e2abx)2φ1(b, qb/eqb/a;eqabx)(44)5 ProofsProof of (25)Take (i; j, k; l, m, n) = (3; 0, 2; 1, 4, 5) in (12) and use F (3) = F (3; 4, 5), to get(β43, β53, α012; q)∞3φ2(α045, α145, α245β43, β53;α012)=(α124, α125, α245, β03; q)∞(β20/q; q)∞3φ2(α014, α015, α045β03, β02; q)+(α014, α015, α045, β23; q)∞(β02/q; q)∞3φ2(α124, α125, α245β23, β20; q).In terms of the parameters a, b, c, e, f and q, this is3φ2(qabc/ef, a, bqab/f, qab/e;qc)=(e/c, f/c, b, q2ab/ef ; q)∞(q/c, qab/f, qab/e, ef/qac; q)∞3φ2(qa/f, qa/e, qabc/efq2ab/ef, q2ac/ef; q)+(qa/f, qa/e, qabc/ef, qb/c; q)∞(q/c, qab/f, qab/e, qac/ef ; q)∞3φ2(e/c, f/c, bqb/c, ef/ac; q).Set f = abcx/e to get3φ2(q/x, a, bqe/cx, qab/e;qc)=(b, abx/e; q)∞(qab/e, bx/q; q)∞(e/c, q2/cx; q)∞(q/c, qe/cx; q)∞3φ2(qe/bcx, qa/e, q/xq2/cx, q2/bx; q)+(qa/e, q/x; q)∞(qab/e, q/bx; q)∞(qb/c, qe/bcx; q)∞(q/c, qe/cx; q)∞3φ2(e/c, abx/e, bqb/c, bx; q). (45)8 R.L.I.M.S. Vol. 2, May 2001Next, observe that as c→ 0,3φ2(q/x, a, bqe/cx, qab/e;qc)→ 3φ1(q/x, a, bqab/e;xe)(46)3φ2(qe/bcx, qa/e, q/xq2/cx, q2/bx; q)→ 2φ1(qa/e, q/xq2/bx;eb)=(qe/abx, qa/b; q)∞(q2/bx, e/b; q)∞2φ1(a, qa/eqa/b;qeabx)(47)3φ2(e/c, abx/e, bqb/c, bx; q)→ 2φ1(abx/e, bbx;eb)=(x, e; q)∞(bx, e/b; q)∞2φ1(a, be;x). (48)The iterate of Heine’s transformation (3) has been used to transform the 2φ1 functions in (47) and(48). Furthermore, using (1) we have(e/c, q2/cx; q)∞(q/c, qe/cx; q)∞= 2φ1(q/e, x/qq/c;qecx)→ 2φ0(q/e, x/q− ;ex)(49)and similarly(qb/c, qe/bcx; q)∞(q/c, qe/cx; q)∞= 2φ1(1/b, bx/eq/c;qecx)→ 2φ0(1/b, bx/e− ;ex). (50)Equation (25) now follows by taking the limit as c→ 0 in (45) and using equations (46) – (50).Remaining proofsThe remaining equations can be proved in the same way. Equations (26) – (32) can also be ob-tained from (25) just by change of variable. A summary of the proofs is as follows.S. Cooper, On the q-analogue of Kummer’s 24 solutions 9Eqn. (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→ abx/e in (25)(28) (3; 2, 4; 0, 1, 5) a→ qa/e, b→ qb/e, e→ q2/e in (25)(29) (5; 0, 2; 1, 3, 4) a→ e/b, b→ e/a, x→ abx/e in (25)(30) (3; 1, 4; 0, 2, 5) a→ qb/e, b→ qa/e, e→ q2/e in (25)(31) (5; 1, 4; 0, 2, 3) a→ q/a, b→ q/b, e→ q2/e, x→ abx/e in (25)(32) (5; 2, 4; 0, 1, 3) a→ q/b, b→ q/a, e→ q2/e, x→ abx/e 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/e, b→ qb/e, e→ q2/e in (33)(42) (4; 1, 2; 0, 3, 5) a→ qa/e, b→ qb/e, e→ q2/e in (34)(43) (5; 0, 4; 1, 2, 3) a→ e/a, b→ e/b, x→ abx/e in (35)(44) (5; 1, 2; 0, 3, 4) a→ e/a, b→ e/b, x→ abx/e in (36)6 RemarksEquation (33) is due to Sears [6, (p. 178 II(a))]. It also can be obtained by taking θr = (abx/ep)rin [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 in [6, (p. 178 II(c))] and [4, (p.92, ex. 3.8)]; the equivalence follows immediately using the transformation formula (7).Equation (35) is due to Sears [6, (p. 178 III(c))]. An equivalent form (use Heine’s transfor-mation (2)) is given in [4, (p. 64, eq. (3.3.5))].In equation (39), ﬁrst interchange a and b, then replace x with x/b and let b→∞, to get(qa/e)∞(q/e)∞(x/e, qe/x)∞(ax/e, qe/ax)∞1φ1(ae;x)+(a)∞(e/q)∞(x/q, q2/x)∞(ax/e, qe/ax)∞1φ1(qa/eq2/e;x)= 2φ1(a, qa/e0;qeax)(51)This formula is a q-analogue of [1, (p. 192, eq. (4.1.13))]. Note however that the right hand side ofequation (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 and R. Roy, Special Functions, (Cambridge University Press, 1999).[2] W. N. Bailey, Generalized Hypergeometric Series, (Cambridge University Press, 1935).10 R.L.I.M.S. Vol. 2, May 2001[3] A. Erdelyi, ed. Higher Transcendental Functions, Vol. 1, (McGraw-Hill, 1953; reprinted RobertE. Krieger Publishing Company, Inc., 1981).[4] G. Gasper and M. Rahman, Basic Hypergeometric Series, (Cambridge University Press, 1990).[5] N. A. Hall, An algebraic identity, J. Lond. Math. Soc. 11, (1936), 276.[6] D. B. Sears, On the transformation theory of basic hypergeometric functions, Proc. Lond.Math. Soc. (2) 53, (1951), 158 – 180.[7] G. N. Watson, The Continuations of Functions Deﬁned by Generalised Hypergeometric Series,Trans. Camb. Phil. Soc. XXI, (1910), 281 – 299.

On the q-analogue of Kummer’s 24 solutions

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On the q-analogue of Kummer’s 24 solutions

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