Research Article A Note on Four-Variable Reciprocity Theorem

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1 Hidwi Publishig Corportio Itertiol Jourl of Mthetics d Mtheticl Scieces Volue 2009, Article ID , 9 pges doi: /2009/ Reserch Article A Note o Four-Vrible Reciprocity Theore Chdrshekr Adig d P. S. Guruprsd Deprtet of Studies i Mthetics, Uiversity of Mysore, Msggotri, Mysore , Idi Correspodece should be ddressed to Chdrshekr Adig, dig c@yhoo.co Received 12 Septeber 2009; Accepted 7 Deceber 2009 Recoeded by Teodor Bulbocă We give ew proof of four-vrible reciprocity theore usig Heie s trsfortio, Wtso s trsfortio, d Ruj s 1 ψ 1 -sutio forul. We lso obti geerliztio of Jcobi s triple product idetity. Copyright q 2009 C. Adig d P. S. Guruprsd. This is ope ccess rticle distributed uder the Cretive Coos Attributio Licese, which perits urestricted use, distributio, d reproductio i y ediu, provided the origil work is properly cited. 1. Itroductio Throughout the pper, we let q < 1 d we eploy the stdrd ottio: 0 : ; q 1, 0 : ; q 1 q, 0 : ; q ; q q ; q, <<. 1.1 Ruj 1 stted severl q-series idetities i his lost otebook. Oe of the beutiful idetities is the two-vrible reciprocity theore. Theore 1.1 see 2. For b / 0, 1 ρ, b ρb, b 1 q/b bq/ q, 1.2 q bq

2 2 Itertiol Jourl of Mthetics d Mtheticl Scieces where ρ, b : 1 1 b 0 1 q 1/2 b q. 1.3 I the recet pst y ew proofs of 1.2 hve bee foud. The first proof of 1.2 ws give by Adrews 3 usig four-free-vrible idetity d Jcobi s triple product idetity. Further, Adrews 4 pplied 1.2 i provig Euler prtitio idetity logues stted i 1. Soshekr d Fthi 5 estblished equivlet versio of 1.2 usig Ruj s 1 ψ 1 sutio forul 6 d Heie s trsfortio 7, 8. Berdt et l. 9 lso derived 1.2 usig the se bove etioed two trsfortios. I fct, Berdt et l. 9 i the se pper hve give two ore proofs of 1.2 oe eployig the Rogers-Fie idetity 10 d the other is purely cobitoril. Usig the q-bioil theore: q 0 t t t, t < 1, q < 1, 1.4 Ki et l. 11 gve uch differet proof of 1.2. Guruprsd d Prdeep 12 lso devised proof of 1.2 usig the q-bioil theore. Adig d Aith 13 estblished 1.2 log the lies of Isil s proof 14 of Ruj s 1 ψ 1 sutio forul. Further, they showed tht the reciprocity theore 1.2 leds to q-itegrl extesio of the clssicl g fuctio. Kg 2 costructed proof of 1.2 log the lies of Vektchliegr s proof of the Ruj 1 ψ 1 sutio forul 6, 15. I 2 Kg proved the followig three- d four-vrible geerliztios of 1.2. For c < < 1d c < b < 1, 1 ρ 3, b, c ρ 3 b,, c b 1 c q/b bq/ q c/ c/b q bq, 1.5 where ρ 3, b, c : 1 1 b 0 c 1 q 1/2 b q c/b,, c 1 b / q, 1.6 d for c, d <, b < 1, ρ 4, b, c, d ρ 4 b,, c, d 1 b 1 d c cd/b q/b bq/ q, d/ d/b c/ c/b q bq 1.7

3 Itertiol Jourl of Mthetics d Mtheticl Scieces 3 where ρ 4, b, c, d : 1 1 b 0 d c cd/b 1 cdq 2 /b 1 q 1/2 b q c/b 1 d/b,, c 1 b, d b / q. 1.8 Kg 2 estblished 1.5 o eployig Ruj s 1 ψ 1 sutio forul d Jckso s trsfortio of 2 φ 1 d 2 φ 2 -series. Recetly 1.5 ws derived by Adig d Guruprsd 16 usig q-bioil theore d Guss sutio forul. Soshekr d Mt 17, 18 obtied 1.5 usig the two-vrible reciprocity theore 1.2, Jckso s trsfortio, d gi two-vrible reciprocity theore by preter ugettio. Zhg 19 lso estblished 1.5. Kg 2 estblished 1.7 o eployig Adrews s geerliztio of 1 ψ 1 sutio forul, Sers s trsfortio of 3 φ 2 -series, d liitig cse of Wtso s trsfortio for teritig very well-poised 8 φ 7 -series 8: α β γ δ ɛ 1 αq 2 q 3/2 α 2 αq/β αq/γ αq/δ αq/ɛ q 1 α βγδɛ αq αq/δɛ δ ɛ αq/βγ αq. αq/δ αq/ɛ αq/β αq/γ q δɛ Recetly M 20, 21 proved six-vrible geerliztio d five-vrible geerliztio of 1.2. The i purpose of this pper is to provide ew proof of 1.7 usig 1.9, Heie s trsfortio: α β γ/β z q γ γ 0 βz z αβz/γ β βz q 0 γ, β q < 1, z < 1, γ < β < d Ruj s 1 ψ 1 sutio forul: 1ψ 1 ; b; z : z b/ z q/z q b q/ b/z b z, q b < 1, < z < Jcobi s triple product idetity sttes tht q 1/2 z q zq 1 z, z/ 0, q <

4 4 Itertiol Jourl of Mthetics d Mtheticl Scieces Adrews 22 gve proof of 1.12 usig Euler idetities. Cobitoril proofs of Jcobi s triple product idetity were give by Wright 23, Chee 24, d Sudler 25. We c lso fid proof of 1.12 i 26.Usig1.12, Hirschhor 27, 28 estblished Jcobi s two-squre d four-squre theores. Soshekr d Fthi 5 d Ki et l. 11 estblished 1 b q 1/2 1 1 b 1 q 1/2 1 b q/b b/ q b Note tht 1.13 which is equivlet to 1.2 y be cosidered s two-vrible geerliztio of Corteel d Lovejoy 29, equtio 1.5 hve give bijective proof of 1.13 usig represettios of over prtitios. All the reciprocity theores 1.2, 1.5,d 1.7 re geerliztios of Jcobi s triple product idetity We lso obti geerliztio of Jcobi s triple product idetity 1.12 which is due to Kg Proof of 1.7 The Four-Vrible Reciprocity Theore O eployig q-bioil theore, we hve cq dq dq q q bq bq 0 dq bq 0 0 cq bq 1 q dq 1 b/d q q cq dq q 0 q O usig Heie s trsfortio 1.10 with α cq, β q, γ q, z q 1, we hve cq q 0 q 1 q 2 q 1 q 0 q 1 1 cq/ 1 1 cq 2 / q 2 cq/ q [ q 1 1 ] cq/ cq/ cq/ q q q 1 cq cq/ q 1 0 q 1 1 cq/ 1 0 cq/. q 2.2

5 Itertiol Jourl of Mthetics d Mtheticl Scieces 5 Substitutig this i 2.1, weobti Now, cq dq dq q cq 0 q bq bq b/d dq q 0 cq/ dq b/d dq/ cq/. bq cq/ q 1 1 dq bq dq bq 1 1 dq bq 0 0 b/d dq/ cq/ cq/ q 1 1 b/d dq bq q 1 cq 1 / /d dq/ 0 cq 2 / b/d dq 1 cq 1 q 1 cq 1 / / b/c cq 1 / 1 dq/ 0 dq2 /, o usig 1.10 with α bq cq2,βq, γ d,z dq dq bq 1 1 dq bq 0 0 b/c cq/ dq/ 1 b/c cq/ dq/ 1 1 b/c dq bq dq/ 0 Substitutig 2.4 i 2.3, weobti 1 1 cq 0 b/d dq 1 q bq 1 dq 1. cq dq dq q cq 0 q bq bq q b/d cq/ b/c dq bq dq/ dq 0 cq bq q b/c dq/ 1 1 b/c dq bq dq/ dq cq cq cq. 2.5 Here, we used 1.10 with α b/d, β q, γ cq 2 /, z dq/.

6 6 Itertiol Jourl of Mthetics d Mtheticl Scieces Chgig c to c/q, d to d/q i 2.5, weget c d q d c bq/c 0 q bq bq c q d/ bq/c d 0 bq d/ c. 1 Iterchgig d b i 2.6, we hve c d q d c q/c 0 q bq b bq c q d/b 0 1 b b 1 q/c d 0 q d/b c. 1 b Subtrctig 2.6 fro 2.7, we deduce tht [ d c 1 q/c c 1 bq/c c ] bq q b d/b 0 1 b d/ b 1 q/c d 0 q d/b c 1 1 bq/c d 1 b 0 bq d/ c Now chge to b/d, b to c/,dz to d/ i 1.11 to obti b/d d c/ 1 q/c dq/b 0 c cd/b b/ q/b q. 2.9 b c/ c/b d/ dq/b Chgig to 1 i the first sutio of the bove idetity d the ultiplyig both sides by 1 d/b 1,wefidtht 1 b/d 1 d 1 q/c c 1 d/b c/ 0 1 d/b 0 1 b 1 b cd/b bq/ q/b q. c/ c/b d/ d/b 2.10

7 Itertiol Jourl of Mthetics d Mtheticl Scieces 7 Usig 1.10 with α bq/c, β q, γ dq/, dz c/ i the first sutio of the bove idetity d the ultiplyig both sides by 1/b,weget 1 q/c c 1 bq/c c b d/b 0 1 b d/ b 1 cd/b bq/ q/b q. c/ c/b d/ d/b 2.11 Substitutig 2.11 i 2.8,weseetht 1 b 1 cd/b c d bq/ q/b q q bq c/ c/b d/ d/b 1 b 1 q/c d 0 q d/b c 1 1 bq/c d 1 b 0 bq d/ c Now settig α cd/b, β cd/b, γ c, δ q, dɛ d i 1.9 d the ultiplyig both sides by 1/1 d/b1 c/b,weobti 0 Iterchgig d b i 2.13, we hve cd/b c d 1 cdq 2 /b q 1/2 1 b q c/b 1 d/b q/c d 0 q d/b c. 1 b 0 cd/b c d 1 cdq 2 / q 1/2 1 b bq c/ 1 d/ 1 bq/c d 0 bq d/ c Substitutig 2.13 d 2.14 i 2.12, we deduce 1.7. Theore 2.1 A four-vrible geerliztio of Jcobi s triple product idetity. For c, d <, b < 1, cd/b c d b/ q/b q b c/ c/b d/ d/b d cq / 1 b q 1/ d/b 1 d cq /b 1 1 b 1 q 1/2 b 1 d/

8 8 Itertiol Jourl of Mthetics d Mtheticl Scieces Proof. Eployig q c bq c q 1/2 cq, c b q 1/2 cq c b 2.16 i the right side of 2.12 d the ultiplyig both sides by b/1 1 b, weobti Ackowledget The uthors thk the oyous referee for severl helpful coets. Refereces 1 S. Ruj, The Lost Notebook d Other Upublished Ppers, Spriger, Berli, Gery, S.-Y. Kg, Geerliztios of Ruj s reciprocity theore d their pplictios, Jourl of the Lodo Mtheticl Society, vol. 75, o. 1, pp , G. E. Adrews, Ruj s lost ote book I: prtil θ fuctios, Advces i Mthetics, vol. 41, pp , G. E. Adrews, Ruj s lost otebook. V. Euler s prtitio idetity, Advces i Mthetics, vol. 61, o. 2, pp , D. D. Soshekr d S. N. Fthi, A iterestig geerliztio of Jcobi s triple product idetity, Fr Est Jourl of Mtheticl Scieces, vol. 9, o. 3, pp , B. C. Berdt, Ruj s Notebooks. Prt III, Spriger, New York, NY, USA, G. E. Adrews, R. Askey, d R. Roy, Specil Fuctios, vol. 71 of Ecyclopedi of Mthetics d Its Applictios, Cbridge Uiversity Press, Cbridge, UK, G. Gsper d M. Rh, Bsic Hypergeoetric Series, vol. 96 of Ecyclopedi of Mthetics d Its Applictios, Cbridge Uiversity Press, Cbridge, UK, 2d editio, B. C. Berdt, S. H. Ch, B. P. Yep, d A. J. Yee, A reciprocity theore for certi q-series foud i Ruj s lost otebook, Ruj Jourl, vol. 13, o. 1 3, pp , N. J. Fie, Bsic Hypergeoetric Series d Applictios, vol. 27 of Mtheticl Surveys d Moogrphs, Aeric Mtheticl Society, Providece, RI, USA, T. Ki, D. D. Soshekr, d S. N. Fthi, O geerliztio of Jcobi s triple product idetity d its pplictios, Advced Studies i Coteporry Mthetics, vol. 9, o. 2, pp , P. S. Guruprsd d N. Prdeep, A siple proof of Ruj s reciprocity theore, Proceedigs of the Jgjeo Mtheticl Society, vol. 9, o. 2, pp , C. Adig d N. Aith, O reciprocity theore of Ruj, Tsui Oxford Jourl of Mtheticl Scieces, vol. 22, o. 1, pp. 9 15, M. E. H. Isil, A siple proof of Ruj s 1 ψ 1 su, Proceedigs of the Aeric Mtheticl Society, vol. 63, o. 1, pp , C. Adig, B. C. Berdt, S. Bhrgv, d G. N. Wtso, Chpter 16 of Ruj s secod otebook: thet-fuctios d q-series, Meoirs of the Aeric Mtheticl Society, vol. 53, o. 315, C. Adig d P. S. Guruprsd, O three vrible reciprocity theore, South Est Asi Jourl of Mthetics d Mtheticl Scieces, vol. 6, o. 2, pp , D. D. Soshekr d D. Mt, A ote o three vrible reciprocity theore, to pper i Itertiol Jourl of Pure d Applied Mthetics, D. D. Soshekr d D. Mt, O the three vrible reciprocity theore d its pplictios, to pper i The Austrli Jourl of Mtheticl Alysis d Applictios. 19 Z. Zhg, A idetity relted to Ruj s d its pplictios, to pper i Idi Jourl of Pure d Applied Mthetics.

9 Itertiol Jourl of Mthetics d Mtheticl Scieces 9 20 X. R. M, Six-vrible geerliztio of Ruj s reciprocity theore d its vrits, Jourl of Mtheticl Alysis d Applictios, vol. 353, o. 1, pp , X. R. M, A five-vrible geerliztio of Ruj s reciprocity theore d its pplictios, to pper. 22 G. E. Adrews, A siple proof of Jcobi s triple product idetity, Proceedigs of the Aeric Mtheticl Society, vol. 16, pp , E. M. Wright, A euertive proof of idetity of Jcobi, Jourl of the Lodo Mtheticl Society, vol. 40, pp , M. S. Chee, Vector prtitios d cobitoril idetities, Mthetics of Coputtio, vol. 18, pp , C. Sudler Jr., Two euertive proofs of idetity of Jcobi, Proceedigs of the Ediburgh Mtheticl Society, vol. 15, pp , G. H. Hrdy d E. M. Wright, A Itroductio to the Theory of Nubers, Oxford Uiversity Press, New York, NY, USA, 5th editio, M. D. Hirschhor, A siple proof of Jcobi s two-squre theore, The Aeric Mtheticl Mothly, vol. 92, o. 8, pp , M. D. Hirschhor, A siple proof of Jcobi s four-squre theore, Proceedigs of the Aeric Mtheticl Society, vol. 101, o. 3, pp , S. Corteel d J. Lovejoy, Overprtitios, Trsctios of the Aeric Mtheticl Society, vol. 356, o. 4, pp , 2004.

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