A RECIPROCITY RELATION FOR WP-BAILEY PAIRS

Transcription

1 A RECIPROCITY RELATION FOR WP-BAILEY PAIRS JAMES MC LAUGHLIN AND PETER ZIMMER Abstract We derive a ew geeral trasformatio for WP-Bailey pairs by cosiderig the a certai limitig case of a WP-Bailey chai previously foud by the authors, ad examie several cosequeces of this ew trasformatio These cosequeces iclude ew summatio formulae ivolvig WP-Bailey pairs Other cosequeces iclude ew proofs of some classical idetities due to Jacobi, Ramauja ad others, ad ideed exted these idetities to idetities ivolvig particular specializatios of arbitrary WP-Bailey pairs 1 Itroductio I the preset paper, we derive a ew geeral trasformatio for WP- Bailey pairs by cosiderig the a certai limitig case of a WP-Bailey chai previously foud by the authors, ad examie several cosequeces of this ew trasformatio These cosequeces iclude ew expressios for various theta fuctios ad ew summatio formulae ivolvig WP-Bailey pairs A WP-Bailey pair see Adrews [1] is a pair of sequeces α a,, q, β a,, q satisfyig α 0 a,, q β 0 a,, q 1, ad for > 0, 11 β a,, q j0 /a; q j ; q j q; q j aq; q j α j a,, q If the cotext is clear, we occasioally suppress the depedece o some or all of a, ad q For a WP-Bailey pair α a,, β a,, defie 12 F a,, q : 1 q 2 q; q 2 1 a ; q qa; q2 qa 1 β a, qa, q; q 2 q a ; q2 q 2 ; q a ; q2 q 2 a 2 2, q 2 a; q 2 qa 2 α2 a, Date: April 15, Mathematics Subject Classificatio Primary: 33D15 Secodary:11B65, 05A19 Key words ad phrases Bailey pairs, WP-Bailey Chais, WP-Bailey pairs, Lambert Series, Basic Hypergeometric Series, q-series, theta series, q-products 1

2 2 JAMES MC LAUGHLIN AND PETER ZIMMER 2 a, q3 a 2 2, q 3 a, q 2 ; q 2 2 q a, q2 a 2 2, q 2 a, q; q q a, q; q2 q 3 a 2 2, q 3 a; q 2 qa 21 α21 a, The mai result of the paper is the followig reciprocity result for the fuctio F a,, q Theorem 1 Let a ad be o-zero complex umbers ad q a complex umber such that q < max{1, a/, /a } ad oe of the deomiators followig vaish Let α a,, β a, be a WP-Bailey pair Let F a,, q be as at 12 The 1 13 F a,, q F a, 1, q aq aq, q/a, 2 /a, q 2 a/ 2, 2 /q, q 3 / 2, q 2, q 2 ; q 2 a 2 aq/ 2, 2 q/a, a, q 2 /a, 2, q 2 / 2, q, q; q 2 2 /a, qa/ 2, a, q/a; q q2, q 2 ; q 2 a 2 a 2 2, q 2 / 2, a 2 / 2, q 2 2 /a 2 ; q 2 1 a1 a 2 2 Implicatios of this result iclude ew represetatios for various theta fuctios Oe example is cotaied i the followig idetity Recall that 14 ψq q 1/2 q2 ; q 2 q; q 2 0 is Ramauja s theta fuctio Let ω : exp2πi/3, let χ 0 deote the pricipal character modulo 3, ad let ψq be as defied at 14 The q 9 χ 0 1 q 2 ψ6 q ψ 2 q 3 ψ3 q 1/2 ψ 3 q 1/2 ψq 3/2 ψq 3/2 31 ω 2 1 ωq 2 q; q 1 ω 2 q; q 2 ωq 1 q 3 ωq; q q; q 2 1 ω 2 q 2 q; q 1 ωq; q 2 ω 2 q 31 ω 1 q 3 ω 2 q; q q; q 2 Aother implicatio is the followig trasformatio for WP-Bailey pairs If α a,, q, β a,, q is a WP-Bailey pair, the 1 q 2 q; q 2 1 q ; q q2 ; q 2 q 2 β 1 q 2 q,, q, q; q 2 ; q 2 q 2 ; q q ; q2 q4 α 2 q,, q q 4 2, q 3 ; q q q q 2 q 2

3 A RECIPROCITY RELATION FOR WP-BAILEY PAIRS q, q/ 2, q 2, q 2 ; q 2 2, q 2 / 2, q, q; q q 2, 1 q ; q2 1 q44 α 21 q,, q q 3 2, q 2 ; q Several other implicatios are to be foud elsewhere i the paper 2 bacgroud The subject of basic hypergeometric series too a leap forward after Adrews developmet i [1] of a WP-Bailey Chai, a mechaism for derivig ew WP-bailey pairs from existig pairs I [1], Adrews i fact describes two such chais Waraar [15] foud four additioal chais ad Liu ad Ma [8] itroduced the idea of a geeral WP-Bailey chai, ad discovered oe ew specific WP-Bailey chai I [13], the authors foud three ew WP-Bailey chais Each WP-Bailey chai also implies a trasformatio coectig the terms i a arbitrary WP-Bailey pair see 31 below for a example of such a trasformatio, leadig to ew trasformatio formulae for basic hypergeometric series I [9], the first author of the preset paper wet i a somewhat differet directio ad foud two ew types of trasformatios for WP-Bailey pairs Theorem 2 McL, [9] If α a,, β a, is a WP-Bailey pair, the subject to suitable covergece coditios, q, q, z; q q; q 1 qa 21, β a,, q, q z ; q z 1 q, q 1, 1 z ; q q; q 1 qz 1 β 1, 1, q, qz ; q a a, 1 z; q q; q 1 qa 1 z qa, qa z ; q α a, ; q q; q 1 qz 1 z q a, qz a ; q α a a, 1 a 1 1 z 1 a z 1 a1 1 a z z, q z, a, qa, a z, qz a, q, q; q z 1 z z, q z, z a, qa z, a, q a,, q ; q Theorem 3 McL, [9] If α a,, q, β a,, q is a WP-Bailey pair, the subject to suitable covergece coditios, 22 1 q 2 z; q q; q 1 qa β a,, q 1 q, q/z; q z 1 q 2 z; q q; q 1 qa β a,, q 1 q, q/z; q z

4 4 JAMES MC LAUGHLIN AND PETER ZIMMER q 4 z 2 ; q 2 q 2 ; q 2 1 q 2 a q 2 2, q 2 2 /z 2 ; q 2 z 2 β a 2, 2, q 2 z; q q; q 1 qa α a,, q qa, qa/z; q z z; q q; q 1 qa α a,, q qa, qa/z; q z z 2 ; q 2 q 2 ; q 2 1 q 2 a 2 2 q 2 a 2, q 2 a 2 /z 2 ; q 2 z 2 α a 2, 2, q 2 Some similar results were obtaied i two other papers, [10, 11], ad various cosequeces of the trasformatios foud were also examied The results i the preset paper may be viewed as derivig from a cotiuatio of the ivestigatios i the papers alluded to above 3 Proof of the mai idetity Before comig to the proofs, we recall a idetity derived by the preset authors i [12] Theorem 4 [12, Mc Laughli, Zimmer] If α a,, β a, is a WP- Bailey pair, the 31 by qab/, q/b; q q, qa/; q q, q, 2 /ab, b, qa, qa; q qa,, qab/, q/b, q/a, β q/a; q 0 q 2, q2 a b ; q2 ab, bq, q2 a 2 b 2 q, 2 q a, q2 a, q2 a 2 ; q ab, b, q3 a 2 b 2, q3 a b ; q2 2 ab, b; q2 q 2 a 2 b 0, q2 a 2 b ; q2 2 q ab, bq; q2 qa 2 α 2 q, 2 q a, q2 a, q2 a 2 ; q 2 q 3 a 2 b 2 0, q3 a 2 b ; q2 We also recall a result from [9], amely that if fa,, z, q is as defied 32 q, q,, z, /a; q qa fa,, z, q,, q, q/z, qa; q 1 q z q a, q a, a, z, a/; q q a, a, qa, qa/z, q; q 1 q z q 1 q q a/z 1 q a/z qa 21 α 21 aq 1 aq q /z 1 q /z,

5 the 33 fa,, z, q f A RECIPROCITY RELATION FOR WP-BAILEY PAIRS 5 1 a, 1, 1 z, q z a 1 1/z1 a/z 1 a1 1 a/z1 /z z, q/z, /a, qa/, a/z, qz/a, q, q; q z/, q/z, z/a, qa/z, a, q/a,, q/; q We remar that the idetity 33 was also proved by the authors i [3], for the fial form of fa,, z, q above We ote two special cases for the proof below Firstly, 34 fa/,, 1, q 2 35 fa/,, 1, q f /a, 1/, 1, q a/ 1 a 2 1 a 2 / a Secodly, 36 f a, 2, aq, q 2 2 q q 2 q 1 2 q 2 2 aq / 1 a 2 q 2 / 2 2 /a, qa/ 2, a, q/a; q q 2, q 2 ; q 2 2, q 2 / 2, a 2 / 2, q 2 2 /a 2 ; q 2 q 2 /q 1 q 2 /q aq 2 1 aq 2 37 f a, 2, aq, q 2 1 f a, 1 2, 1 aq, q2 a aq 1 2 q 1 a q 1 2 aq 2 aq q2 1 2 aq q2, aq 2 aq, q/a, 2 /a, q 2 a/ 2, 2 /q, q 3 / 2, q 2, q 2 ; q 2 aq/ 2, 2 q/a, q, q, a, q 2 /a, 2, q 2 / 2 ; q 2 Proof of Theorem 1 Rewrite 31 as 38 q 2 qab/, q/b; q q, qa/; q 1 q 2 2 /ab, b; q qa; q 2 qa 1 qab/, q/b; q 2 q/a; q 2 β a, ab, bq, q2 a 2 b, q2 a 2 b ; q2 q, 2 q a, q2 a, q2 a 2 ; q ab, b; q2 qa 2 α2 a, q 2 a 2 b, q2 a 2 b ; q2

6 6 JAMES MC LAUGHLIN AND PETER ZIMMER 2 ab, b, q3 a 2 b 2 q, 2 q a, q2 a, q2 a 2, q3 a b ; q2 2 ; q 2 0 q 2 2 q ab, bq; q2 q 3 a 2 b, q3 a 2 b ; q2 qa 21 α21 a, ab, bq, q2 a 2 b, q2 a 2 b ; q2 qab/, q/b; q q, 2 q a, q2 a, q2 a 2 ; q 2 q, qa/; q 2 If we divide through by 1 b ad the let b 1 o the left side of 38, the the result is F a,, q If we defie Hb : qab/, q/b; q q, 2 q a, q2 a, q2 a 2 ; q 2 2 q, qa/; q q 2 ab, bq, q2 a 2 b, q2 a 2 b ; q2 the we see that dividig the right side of 38 by 1 b gives q 2, q2 a b ; q2 ab, bq, q2 a 2 b 2 q, 2 q a, q2 a, q2 a 2 ; q 2 2 so that the result of lettig b 1 is 39 H 1 q 1 q aq / 1 aq / a 2 q 2 / 2 1 a 2 q 2 / 2 q 2 /q 1 q 2 /q q 1 2 q 2 aq / 1 a 2 q 2 / 2 2 q q 2 1 Hb, 1 b q 2 /q 1 q 2 /q 1 2 fa/,, 1, q f a, 2, aq, q 2 aq 2 1 aq 2 aq 2 1 aq 2 2 aq q2 1 2 aq q2 2 aq q2 1 2 aq q2 Here the first equality is by logarithmic differetiatio otig that H1 1, the secod is by simple combiatio/separatio of some of the series, ad the fial equality follows from 34 ad 36 Thus we have that 310 F a,, q 1 2 f a,, 1, q f a, 2, aq, q 2 Upo replacig a with 1/a ad with 1/ i F a,, q ad the Lambert series above, we get 1 F a, 1, q 12 a f, 1 1, 1, q f a, 1 2, 1 aq, q2 q 2q 1 q a 1 2 q a

7 Thus, F a,, q F A RECIPROCITY RELATION FOR WP-BAILEY PAIRS 7 1 a, 1, q 12 a f,, 1, q 12 a f, 1, 1, q f a, 2, aq, q 2 1 f a, 1 2, 1 aq, q2 q 2q 1 q a 1 2 q ad 12 follows from 35 ad 37, upo otig that a/ 1 a a a 2 / aq 1 2 q 1 a q 1 2 aq q 2q 1 q a 1 2 q Oe easy implicatio is the followig summatio formula a a 2 a 2 1 a1 a 2 2 Corollary 1 Let a ad be o-zero complex umbers ad q a complex umber such that q < max{1, a/, /a } ad suppose oe of the deomiators followig vaish The 1 q 2 q; q 1 2 /a,, /a; q 311 qa; q2 qa 1 qa/, q, aq, q; q 2 q/a; q 2 1 q 2 /q; q 1 a/ 2, 1/, a/; q q/a; q 2 q 1 1/ q/a, q/, q/a, q; q aq/ 2 ; q 2 a aq aq, q/a, 2 /a, q 2 a/ 2, 2 /q, q 3 / 2, q 2, q 2 ; q 2 2 aq/ 2, 2 q/a, a, q 2 /a, 2, q 2 / 2, q, q; q 2 a 2 /a, qa/ 2, a, q/a; q q2, q 2 ; q 2 a 2 a 2 2, q 2 / 2, a 2 / 2, q 2 2 /a 2 ; q 2 1 a1 a 2 2 Proof Isert the trivial Bailey pair { 1 0, 312 α a, q 0, > 0, β a, q, /a; q aq, q; q a, ito 13

8 8 JAMES MC LAUGHLIN AND PETER ZIMMER Isertig the uit WP-Bailey pair, 313 α a, q a, q a, a, a/; q a, a, q, q; q { 1 0, β a, 0, > 1,, a liewise leads to a four-term summatio formula, with the same right side as New θ-fuctio Idetities ad ew trasformatios for basic hypergeometric series We cosider some other implicatios of 12 ad 13 Corollary 2 Let q < 1 ad α a,, q, β a,, q be a WP-Bailey pair The 41 1 q 2 q; q 2 1 q ; q q2 ; q 2 q 2 β 1 q 2 q,, q, q; q 2 ; q 2 q 2 ; q q ; q2 q4 α 2 q,, q q 4, q 3 ; q q, q/ 2, q 2, q 2 ; q 2 2, q 2 / 2, q, q; q 2 Proof From 39 ad 310, F q,, q 42 q 1 2 q q 1 2 q 2 q /q 1 q 2 /q q q q 2 q 2 q 2, 1 q ; q2 1 q44 α 21 q,, q q 3 2, q 2 ; q q 1 / 1 q 22 / 2 q 1 q / 1 1/ 2 q/ 1 q 2 / 2 q 1 q q 1 2 q 2 2 q, q/ 2, q 2, q 2 ; q 2 2, q 2 / 2, q, q; q 2 q /q 2 1 q 2 /q 2 2 q q q 2 q 2 2 q q q 2 q 2, where a idetity of Ramauja [4, Chapter 17, page 116, Equatio 85] is used to combie the two Lambert series to give the ifiite product Upo usig 12 to substitute for F q,, q above, we get the idetity

9 A RECIPROCITY RELATION FOR WP-BAILEY PAIRS q 2 q; q 2 1 q ; q q2 ; q 2 q 2 β 1 q 2 q,, q; q 2 ; q 2 q 2 ; q q ; q2 q 2 2 α q 4 2q,, q 3 ; q q, q5, q 4, q 2 ; q 2 2 2, q; q 2 q 2 21 α 2, q4, q 3, q; q 2 q 5 21q, 2 0, q 4 ; q q, q/ 2, q 2, q 2 ; q 2 2 q q 2 2, q 2 / 2, q, q; q q 2 q 2 This last idetity gives 41, after some simple maipulatios Remar: We ote i passig that if Rq deotes the left side of 41, ad Sq deotes the ifiite series followig the ifiite product o the right side of 41, the 1 2 Rq Sq is ivariat uder the trasformatio 1/ Let 44 M, q : 2 q, q/ 2, q 2, q 2 ; q 2 2, q 2 / 2, q, q; q 2, the ifiite product at 41 above We also ote that may of the theta fuctios ivestigated by Ramauja ad others are expressible i terms of M, q, so that isertig the WP-Bailey pair 0, if is odd, 45 α a, q 2 a, q 2 a, a, a 2 / 2 ; q 2 /2, if is eve, β a, a, a, q 2, q 2 2 /a; q 2 a /2, q/a, q/a, a/; q aq, aq, q 2 /a, q; q, a with a q ad the appropriate choice which we give below, i 41 provides represetatios of these theta fuctios i terms of basic hypergeometric series For example, recall the fuctio 46 aq : q m2 m 2 m, This series was studied i [6], where it was show that a 3 q b 3 q c 3 q,

10 10 JAMES MC LAUGHLIN AND PETER ZIMMER where bq m, ωm q m2 m 2, ω exp2πi/3, ad cq m, qm1/32 m1/31/31/3 2 The series aq was also studied by Ramauja, who showed Etry 1828 of Ramauja s Lost Noteboo - see [2, page 402] that aq 1 6 q 2 q 3 1 q 2 q 3 6 q 1 q 3 1 q 1 q 3 I [5, Equatio 63, page 116] the author showed that 47 aq aq 2 6q q, q5, q 6, q 6 ; q 6 q 2, q 4, q 3, q 3 ; q 6 6q ψ3 q 3 ψq This result may be derived from 310 by replacig q with q 3, a with 1/q 3 ad with 1/q 2, the isertig the uit WP-Bailey pair 313 ad fially usig 47 to combie the resultig Lambert series Notice that the last idetity gives that aq aq 2 6qMq, q 3, with the implicatios give by 41 oted above As a secod example, cosider the fuctio qψq 2 ψq 6 Ramauja showed see Etry 3 i, Chapter 19, page 223 of [4] that, 48 qψq 2 ψq 6 q 65 1 q 1210 q 61 1 q 122 With regard to 41, we ote that qψq 2 ψq 6 qmq, q 6 Thirdly, recall the fuctio 49 ϕq : q 2 q, q, q 2 ; q 2 Ramauja Etry 8 i i chapter 17 of [4] showed that ϕq 2 q q 43 4 q 41 1 q 41, ad is easy to chec that ϕq 2 2Mi, q, where i A variatio of 41 It is ot difficult to see that if we set a q 2t1, where t is a iteger, i 39 ad 310, the the Lambert series combie i essetially the same way as they did at 42, with a differet fiite sum of terms left over from combiig the Lambert series that essetially cacel each other, the particular sum depedig o the choice of t Corollary 2 follows from the choice t 0, but a similar result will follow from other choices for t If we let t be a egative iteger, the isertig the usual WP- Bailey pairs i the resultig idetity will, i most cases, give trivial results for most WP-Bailey pairs, either the α or the β cotais a a; q factor, or some similar factor that will vaish for all large eough whe a is a egative power of q However, there are two WP-Bailey pairs which give o-trivial results for the choice a q 1, ad we cosider those results ext

11 A RECIPROCITY RELATION FOR WP-BAILEY PAIRS 11 The proof of the followig corollary is virtually idetical to that of Corollary 3 except, as metioed, a is set equal to 1/q ad so is omitted Corollary 3 Let q < 1 ad α a,, q, β a,, q be a WP-Bailey pair Defie β1/q,, q lim a 1/q 1 aqβ a,, q, if the limit exists The, assumig all series coverge, q 2 q; q 1 2 q; q q2 ; q , q; q 2 q 2 ; q 2 2 q, q/ 2, q 2, q 2 ; q 2 2, q 2 / 2, q, q; q 2 Isertig the WP-Bailey pair 411 q 2 ; q q; q 2 α 21/q,, q 1, q; q α 1 a, qa2 / 2 ; q q, q 1 β 1/q,, q 2 q 2, q; q 2 α 211/q,, q q 2, q 2 ; q 2 2, a β 1 a, qa/, ; q 2 /a; q 2 2, /a, q, q aq, q 2 leads, for > 1, to the idetity 1 q 2 2 q; q q, q 1 q q; q 2 2 q ; q2 q2 1 q 2 q, q; q 2 2 q, q/ 2, q 2, q 2 ; q q 2, 1 ; q 2 1 2, q 2 / 2, q, q; q 2 2 q 2 q21 1 q 21 q 2, q 2 ; q 2 0 Similarly, isertig the pair at 45 gives the idetity 1 q 2 q; q 1 2 q,, 1 q ; q q 413 q, 2 q 2, 1, q; q2 1 q 41 q 2 ; q q,, 1 2 q 2 q ; q2 q2 1 q 1 2 q 3, 1, q, q 2 ; q q, q/ 2, q 2, q 2 ; q 2 2, q 2 / 2, q, q; q 2 The ext result also follows from 12, ad expresses a geeral sum ivolvig a arbitrary WP-Bailey pair i terms of Lambert series Corollary 4 Let α a,, β a, be a WP-Bailey pair The

12 12 JAMES MC LAUGHLIN AND PETER ZIMMER 1 q 2 q, q; q 1 q 2 ; q q, q; q q; q 2 q β 2, q 2, q 2 ; q 2 1 q 2 2, q 2 2 ; q 2 q 2 α 2 2, q 3 2, q 3 2, q 2, q 2 ; q 2 q, q; q 2 q 2 2, q 2 2, q, q; q 2 q 3 2, q 3 2 ; q 2 q 21 α 21 2, 0 q 1 q 2 2 q 2 1 q 2 q 1 q Proof Defie 415 G, q : 1 q 2 q, q; q 1 q 2 ; q 2 1 q, q; q q; q 2 q β 2, q 2, q 2 ; q 2 1 q 2 2, q 2 2 ; q 2 q 2 α 2 2, q 3 2, q 3 2, q 2, q 2 ; q 2 q, q; q 2 q 2 2, q 2 2, q, q; q 2 q 3 2, q 3 2 ; q 2 q 21 α 21 2, 0 From 12, it ca be see that F 2,, q 0 ad that 416 F a,, q G, q lim a /a lim af a,, q F 2,, q a 2 a 2 2 M 2, where Ma : F a,, q The result follows from the represetatio of Ma as a sum of Lambert series o the right side of 310, after some simple algebraic maipulatios, ad after usig the idetity xq 1 xq 2 x q 1 q a umber of times The idetity at 414 exteds a result by the first author i a previous paper [11, Equatio 73], where the idetity which follows from 414 upo isertig the trivial WP-Bailey pair 312 was prove Upo replacig q with q 2 ad with 1/q i 414 ad isertig the uit WP-Bailey pair 313, we derive Ramauja s idetity Example iii o page 139 of [4] 417 qψ 4 q q 21 1 q 42 where ψq is defied at 14 The same substitutios i 414 followed by the isertio of the trivial WP-Bailey pair 312 gives the idetity [11,

13 Corollary14] 418 A RECIPROCITY RELATION FOR WP-BAILEY PAIRS q 43 q 2 ; q 2 q 4 ; q 4 q 1 q 21 q 2 ; q 2 1 q 2 ; q 4 1 ψ 4 q 2 It is possible to derive a more geeral idetity ivolvig the fuctio qψ 4 q 2 Corollary 5 Let α a,, q, β a,, q be a WP-Bailey pair The q 2 q, q; q 1 q β 1, 1, q 2 q, q; q 1 q 2 q, q; q 1 q β 1, 1, q 2 q, q; q q 1 q 2 α q 1, 1, q 1 q 2 α 1, 1, q 4qψ 4 q q 2 q; q 1 q 1/2 1 q 2 q; q 1 q 1/2 1 q 2 q; q 1 q 2 2q q; q q 82 4 q 82 3 q 62 q 42 1 q 2 3q 86 1 q 26 1 q q q qψ 4 q 2 Proof From 415, the left side of 419 is G1, q G1, q O the other had, from 414, q G1, q 1 q 2 q 2 1 q 2 1 q 1 q Thus 2 2 q 1 q 2 1q 21 1 q 21 1 q 1 q 2 1q 21 1 q 42 2 G1, q G1, q 4 ad 419 follows from q 42 1 q q 21 1 q 42,

14 14 JAMES MC LAUGHLIN AND PETER ZIMMER For 420, we start with Sigh s WP-Bailey pair [14], α a,, q q a, q a, a, ρ 1, ρ 2, a 2 q/ρ 1 ρ 2 ; q a, a, q, aq/ρ 1, aq/ρ 2, ρ 1 ρ 2 /a; q β a,, q ρ 1/a, ρ 2 /a,, aq/ρ 1 ρ 2 ; q aq/ρ 1, aq/ρ 2, ρ 1 ρ 2 /a, q; q, set a 1, 1 ad let ρ 1, ρ 2 to get the pair α 1, 1, q 1 q 1 q 1/2, β 1, 1, q 1; q q 1/2 q; q, a The first two series i 420 come directly from isertig the expressios for β 1, 1, q ad β 1, 1, q i the first two series i 419 The third series i 420 comes isertig the expressios for α 1, 1, q ad α 1, 1, q i the last two series i 419, the replacig, i tur, with 4, 4 1, 4 2 ad 4 3, ad the combiig each pair of series ito a sigle series, ad fially combiig the three survivig series together ito oe series Remar: It is ot difficult to see that a similar cosideratio of at 13 gives the followig result F a,, q F 1/a, 1/, lim a /a Corollary 6 Let α a,, β a, be a WP-Bailey pair ad let G, q be as defied at 415 The 421 G, q G1/, q 2q 1 q 2 2 q 2 1 q 2 q 2 / 2 1 q 2 q 1 q q, q, 2, q/ 2 ; q q 2, q 2 ; q 2 2, 2, q 2 / 2, q 2 / 2 ; q 2 q / 1 q 2 2 q, 2 q, q/ 2, q/ 2, q 2, q 2, q 2, q 2 ; q 2 2, 2, q 2 / 2, q 2 / 2, q, q, q, q; q 2 The special case of the this idetity that follows from isertig the trivial pair 312 ito the term G, q G1/, q also follows from Corollary 12 i [11], upo dividig the idetity there by 1 b ad the lettig b 1 As well as implyig some of the ow idetities relatig Lambert series ad ifiite products, the term G, q G1/, q i 421 also provides a additioal expressio for the Lambert series or the ifiite product i terms of series ivolvig a arbitrary WP-Bailey pair α a,, β a, with a 2 ad specialized as required We give two examples

15 First recall that A RECIPROCITY RELATION FOR WP-BAILEY PAIRS 15 ϕq : Corollary 7 If q < 1, the q 1 q ϕ4 q q 2 q; q 2 2 q 2 ; q 2 q, q2 ; q 2 q, q 2 ; q iq 2 q, q; q 1 1; q 2 iq 1 iiq, iq; q q; q 2 1 iq 2 q, q; q 1 1; q 2 iq 1 iiq, iq; q q; q 2 Proof Let i i 421 It is ot difficult to show that the Lambert series combie to give 2q q q 21 q 1 q q It is also easy to see that the ifiite product side simplifies to give ϕ4 q For Gi, q G1/i, q, we use 415 with i ad isert the trivial WP- Bailey pair 312 with a 2 ad the i: α 0, > 0, β i, i; q q, q; q 1; q2 q 2 ; q 2 Multiply the resultig set of equalities by 4, add 1, ad the idetities at 422 follow Remars: The first equality at 422 is due to Jacobi [7] Also, if other WP-Bailey pairs are used istead of the trivial pair above, the still further represetatios for ϕ 4 q will result Corollary 8 Let ω : exp2πi/3, let χ 0 deote the pricipal character modulo 3, ad let ψq be as defied at 14 The 423 q 9 χ 0 1 q 2 ψ6 q ψ 2 q 3 ψ3 q 1/2 ψ 3 q 1/2 ψq 3/2 ψq 3/2 31 ω 2 1 ωq 2 q; q 1 ω 2 q; q 2 ωq 1 q 3 ωq; q q; q 2

16 16 JAMES MC LAUGHLIN AND PETER ZIMMER 31 ω 1 ω 2 q 2 q; q 1 ωq; q 2 ω 2 q 1 q 3 ω 2 q; q q; q 2 Proof The proof is similar to that of the previous corollary, except we set ω i 421 The Lambert series combie to give 3 1q q 32 q 3 1 q q 64 3 χ 0 1 q 2 The ifiite product side simplifies to give 1 ψ 6 q 3 ψ 2 q ψ 3 q 1/2 ψ 3 q 1/2 ψq 3/2 ψq 3/2 Oce agai, for Gω, q G1/ω, q we use 415, this time with ω ad the trivial WP-Bailey pair 312 with a 2 ad the ω: α 0, > 0, β ω, ω2 ; q ω 2 q, q; q Multiply the resultig set of equalities by 3, ad the idetities at 423 follow, after some simple maipulatios of the expressios for Gω, q ad Gω 2, q Remar: The Lambert series i the idetity at 423 may also be represeted i terms of the theta series aq defied at46, upo otig that q χ 0 1 q 2 χ 0 q 1 q χ 0 q2 1 q 2, ad employig a idetity of Ramauja from the Lost Noteboo, Etry 1829 see [2], page 402, which states that a 2 q 1 12 χ 0 q 1 q Refereces [1] Adrews, George E Bailey s trasform, lemma, chais ad tree Special fuctios 2000: curret perspective ad future directios Tempe, AZ, 1 22, NATO Sci Ser II Math Phys Chem, 30, Kluwer Acad Publ, Dordrecht, 2001 [2] GE Adrews ad BC Berdt, Ramauja s Lost Noteboo, Part I, Spriger, 2005 [3] Adrews, George E; Lewis, Richard; Liu, Zhi-Guo A idetity relatig a theta fuctio to a sum of Lambert series Bull Lodo Math Soc , o 1, [4] B C Berdt, Ramauja s Noteboos, Part III, Spriger-Verlag, New Yor, 1991 [5] B C Berdt, Ramauja s Noteboos, Part V, Spriger-Verlag, New Yor, 1998 [6] Borwei, J M; Borwei, P B, A cubic couterpart of Jacobi s idetity ad the AGM Tras Amer Math Soc , o 2, [7] C G J Jacobi Fudameta Nova Theoriae Fuctioum Ellipticarum, Sumptibus fratrum Borträgar, Regiomoti, 1829

17 A RECIPROCITY RELATION FOR WP-BAILEY PAIRS 17 [8] Q Liu; X Ma O the Characteristic Equatio of Well-Poised Baily Chais - To appear [9] J Mc Laughli, Some ew Trasformatios for Bailey pairs ad WP-Bailey Pairs Cet Eur J Math , o 3, [10] Mc Laughli, James, Some Further Trasformatios for WP-Bailey Pairs - submitted [11] Mc Laughli, James, New Summatio Formulae for some q-products - submitted [12] Mc Laughli, James; Zimmer, Peter Some idetities betwee basic hypergeometric series derivig from a ew Bailey-type trasformatio J Math Aal Appl , o 2, [13] Mc Laughli, James; Zimmer, Peter Geeral WP-Bailey Chais - To appear i The Ramauja Joural [14] Sigh, U B A ote o a trasformatio of Bailey Quart J Math Oxford Ser , o 177, [15] Waraar, S O Extesios of the well-poised ad elliptic well-poised Bailey lemma Idag Math NS , o 3-4, Mathematics Departmet, 25 Uiversity Aveue, West Chester Uiversity, West Chester, PA address: jmclaughl@wcupaedu Mathematics Departmet, 25 Uiversity Aveue, West Chester Uiversity, West Chester, PA address: pzimmer@wcupaedu