Bailey [1] established a simple but very useful identity: If

Transcription

1 itlin journl of pure nd pplied mthemtics n ( ) 179 CERTAIN TRANSFORMATION AND SUMMATION FORMULAE FOR q-series Remy Y Denis Deprtment of Mthemtics University of Gorkhpur Gorkhpur Indi e-mil: ryddry@gmilcom ddry@rediffmilcom SN Singh Deprtment of Mthemtics TDPG College Junpur-00 Indi SP Singh Deprtment of Mthemtics TDPG College Junpur-00 Indi Abstrct In this pper mking use of certin summtion formule n ttempt hs been mde to estblish certin new nd interesting trnsformtion nd summtion formule for q-series Keywords: summtion trnsformtion q-series AMS Subject Clssifiction: 33A30 1 Introduction Biley 1 estblished simple but very useful identity: If (11) β n n u n r v n+r α r r0 nd (1) γ n u r n v r+n δ r rn

2 180 remy y denis sn singh sp singh where α r δ r u r nd v r re ny functions of r only such tht series γ n exists then subject to the convergence of the series (13) α n γ n β n δ n Mking use of (13) Slter 3 gve long list of Rogers-Rmnujn type identities Lter on number of mthemticins notbly Verm 7 Verm nd Jin 9 Singh 5 Denis Singh 6 nd others mde use of Biley s identity (13) nd estblished number of trnsformtion formule nd lso Rogers-Rmnujn type identities of different moduli In this pper mking use of certin known summtion formule due to Verm nd Jin 9 nd identity (13) n ttempt hs been mde to estblish certin very interesting trnsformtion formule for q-hypergeometric series In the lst section of this pper mking use of the following identity due to Verm 8 viz (14) ( z) n q n(n 1)/ q; q n γq n ; q n k0 α β; q n+k B n+k z k q; q k γq n+1 ; q k A n B n (zw) n q; q n n j0 q n γq n ; q j A j (wq) j q; q j α β; q j nd summtion formule due to Verm nd Jin 9 n ttempt hs been mde to estblish certin new trnsformtion nd summtion formule for bsic hypergeometric series Definitions nd nottions A bsic (q-) hypergeometric series is generlly defined to be series of the type n z n where n+1 / n is rtionl function of q n q being fixed complexprmeter clled the bse of the series usully with modules less thn one An explicit representtion of such series is given by 1 (1) r ; q; z rφ s b 1 b b s ; q i nd q in(n 1)/ 1 r ; q n z n q b 1 b b s ; q n 1 r ; q n 1 ; q n ; q n r ; q n with the q-shifted fctoril defined by { 1 if n 0 () ; q n (1 )(1 q)(1 q ) (1 q n 1 ) if n 1

3 certin trnsformtion nd summtion formule for q-series 181 For convergence of the series (1) we need q < 1 nd z < when i > 0 or mx{ q z } < 1 when i 0 provided no zeros pper in the denomintor Following summtions re needed in our nlysis (3) Φ 1 b ; q ; c/b c c/ c/b; q c c/b; q (Slter ; AppIV(IV)) (4) Φ 1 b ; q ; c/b cq cq/ cq/b; q cq cq/b; q { b(1 + c) c( + b) b c } (Verm 7; (14)) (5) 4Φ 3 q n x y q n+1 x xq ; q ; q xyq xyq x q xn q; q n x q ; q m y q ; q m x q; q n x y q ; q m q ; q m where m is gretest integer n/ (6) 4Φ 3 q n b x 4 y q n+ x x q ; q ; q bx q bx q x 4 q xn q; q n bq; q n x q; q n bx q; q n Verm nd Jin 9; (3)) (7) 4Φ 3 q n bx q n+ x xq ; q ; q xq b xq b x q xn q; q n bxq ; q n bx q 3 ; q m bq ; q m xq ; q m xq; q n bx q ; q n q ; q m x q 3 ; q m bxq ; q m where m is gretest integer < n/ Verm nd Jin 9; (3)) (8) 5Φ 4 q q 3 q 3n+3 q 3n ; q 3 ; q 3 (q) 3/ (q) 3/ 3/ q 3 3/ q 3 n q 3 ; q 3 n q; q n 3 q 3 ; q 3 n q; q n (9) Verm nd Jin 9; (4)) x ωxq ω 5Φ xq x 3 q n+4 q n ; q ; q 4 (xq) 3/ (xq) 3/ x 3/ q x 3/ q xn x q 4 ; q n q; q n x 3 q 6 ; q 3 n xq 3 ; q 3m x 3 q 4 ; q n xq; q n q 3 ; q 3 m x q 4 ; q 3m Verm nd Jin 9; (44))

4 18 remy y denis sn singh sp singh where m is gretest integer n/3 nd ω e πi/3 (10) 5Φ 1/3 ω 1/3 ω 1/3 q n+1 q n ; q ; q 4 q q q ( ) n m q; q n q 3 ; q 3 m q q; q n q 3 ; q 3 m where m is gretest integer n/3 nd ω e πi/3 (11) 6Φ 1/3 ω 1/3 ω 1/3 q q n+1 q n ; q ; q 5 q q q where m is gretest integer n/3 q; q n ; q n q 3 ; q 3 m q 6 ; q 3 m ( ) n m q; q n q ; q n q 3 ; q 3 m ; q 3 m Verm nd Jin 9; (45)) Verm nd Jin 9; (48)) 3 Min Results In this section we shll estblish the trnsformtion formule by mking use of (13) Tking u r 1 q; q r v r q; q r q r / If (31) β n q; q n q; q n q n / nd (3) γ n q; q n q n then (33) α n γ n r0 n r0 in (11) nd (1) we get: ( 1) r q r/ q n ; q r q n+1 ; q r α r q r q n+1 ; q r δ r+n q; q r q r /+nr β n δ n infty provided the series involving re convergent We shll now use (31) (3) nd (33) in order to estblish the required trnsformtions (i) Replcing by x y in (31) nd (3) nd then tking α r x xq; q r q r +r/ ( ) r q xyq xyq x q; q r in (31) nd mking use of (5) we hve:

5 certin trnsformtion nd summtion formule for q-series 183 (34) β n x y q; q n x n x q ; q m y q ; q m q n / x q; q n x y q ; q m q ; q m where m is the gretest integer < n/ Agin tking δ r z r q r / in (3) we get fter some simplifiction: (35) γ n x y zq; q x y q; q n ( ) n q n/ z; q q n x q zq; q n q/z; q n Now putting these vlues of α n β n γ n nd δ n in (33) we get: (36) x y zq; q x xq xyq 4Φ 1/ xyq 1/ ; q ; q 3 z; q x q x y zq q/z x Φ y q y q ; q ; x z 1 x q + xz(1 x y q) (1 x q) Φ 1 x y q 3 y q ; q ; x z x q 3 xz < 1 (ii) Next replcing by b x 4 nd q by q in (31) nd (3) nd then tking x x q; q r q r +r ( ) r α r in (31) nd mking use of (6) we hve: bx q bx q x 4 q ; q r q ; q r (37) β n b x 4 q ; q n x n q; q n bq; q n q ; q n x q; q n bx q; q n q n Aginsetting δ r z r q r in (3) we hve: (38) γ n b x 4 zq ; q b x 4 q ; q n ( ) n q n z; q q n b x 4 zq ; q n q /z; q n Putting these vlues of α n β n γ n nd δ n in (33) we get the following trnsformtion: (39) b x 4 zq ; q x 4Φ x q bx q bx q ; q ; q z; q 3 x 4 q b x 4 zq q /z Φ 1 bx q bq ; q ; x z x q x z < 1 (iii) Agin putting bx q in (31) nd (3) nd then tking x xq; q r ( ) r q r +r/ α r x q xq b xq in (31) nd mking use of (7) we get: b; q r q; q r

6 184 remy y denis sn singh sp singh (310) β n xn bxq ; q n bx q 3 ; q m bq ; q m xq ; q m q n / xq; q n q ; q m x q 3 ; q m bxq ; q m where m is the gretest integer n/ Agin tking δ r z r q r / in (3) we get: (311) γ n bx q z; q bx q ; q n ( ) n q n/ z; q q n bx q z; q n q/z; q n Now putting these vlues in (33) we get the following trnsformtion: (31) bzx q ; q x xq xq bq xq bq ; q ; q 4Φ 3 z; q bzx q x q q/z bx 3 Φ q 3 bq xq 3 ; q ; x z xq x q 3 + xz(1 bxq ) (1 xq) bxq 3Φ 4 bx q 3 bq ; q ; x z x q 3 bxq x z < 1 (iv) Next replcing by 3 nd q by q 3 in (31) nd (3) nd then tking q q ; q 3 r q 3r+3r/ ( ) r α r in (31) nd mking use of q 3 (q) 3/ (q) 3/ 3/ q 3 3/ q 3 ; q 3 r (8) we hve: (313) β n q; q n n q 3n / q; q n Agin tking δ r z r q 3r / in (3) we get fter some simplifictions (314) γ n 3 q 3 ; q 3 n 3 q 3 z; q 3 ( ) n q 3n/ z; q 3 3 q 3 z; q 3 n q 3 /z; q 3 n Substituting these vlues of α n β n γ n nd δ n in (33) we get the following summtion (315) 3Φ q q ; q 3 ; q 3 3 q 3 z q 3 /z z; q3 zq; q 3 q 3 z; q 3 z; q (v) Next replying by x 3 q 3 in (31) nd (3) nd then tking x ωxq ω xq; q r q r +r/ ( ) r α r in (31) nd mking use of q (xq) 3/ (xq) 3/ x 3/ q x 3/ q ; q r (9) we get:

7 certin trnsformtion nd summtion formule for q-series 185 (316) β n xn x q 4 ; q n x 3 q 6 ; q 3 m xq 3 ; q 3m q n / xq; q n q 3 ; q 3 m x q 4 ; q 3m where m is the gretest integer n/3 nd ω e πi/3 Agin tking δ r z r q r / in (3) we get: (317) γ n x3 zq 4 ; q x 3 q 4 ; q n ( ) n q n/ z; q q n x 3 zq 4 ; q n q/z; q n (318) Now putting these vlues of α n β n γ n nd δ n in (33) we hve: x 3 zq 4 ; q x ωxq ω 5Φ xq x 3/ q 5/ x 3/ q 5/ ; q ; q 4 z; q (xq) 3/ (xq) 3/ x 3 zq 4 q/z x 3 Φ q 5 x q 6 x 3 q 6 ; q 3 ; x 3 z 3 xq xq + xz(1 x q 4 ) x 4Φ q 5 x q 6 x q 7 x 3 q 6 ; q 3 ; x 3 z 3 3 (1 xq) xq xq 4 x q 4 + x z (1 x q 4 )(1 x q 5 ) x 4Φ q 6 x q 7 x q 8 x 3 q 6 ; q 3 ; x 3 z 3 (1 xq)(1 xq 3 ) xq 4 xq 5 x q 4 x 3 z 3 < 1 (vi) Tking α r 1/3 ω 1/3 ω 1/3 ; q r q r +r/ ( ) r q q q q; q r of (10) we get: in (31) nd mking use (319) β n ()n m/ q 3 ; q 3 m q n / q 3 ; q 3 m where m is the gretest integer n/3 nd ω e πi/3 Agin tking δ r z r q r / in (3) we get: (30) γ n zq; q q; q n ( ) n q n/ z; q q n zq; q n q/z; q n (31) Now putting these vlues of α n β n γ n nd δ n in (33) we get: 4Φ 3 1/3 ω 1/3 ω 1/3 q ; q ; q zq q/z z; q z 3 q 3 ; q 3 zq; q z 3 ; q 3 {1 + 1/ z + z }

8 186 remy y denis sn singh sp singh (vii) Lstly tking α r 1/3 ω 1/3 ω 1/3 q ; q r ( ) r q r +r/ q q q q q ; q r mking use of (11) we get: in (31) nd (3) β n ; q n q 3 ; q 3 m q 6 ; q 3 m ( ) n m q n / q ; q n q 3 ; q 3 m ; q 3 m where m is the gretest integer n/3 nd ω e πi/3 Agin tking δ r z r q r / in (3) we get: (33) γ n zq; q q; q n ( ) n q n/ z; q q n zq q/z; q n Now putting these vlues in (33) we get: (34) zq; q 6Φ 1/3 ω 1/3 ω 1/3 q q q ; q ; q 5 z; q zq q/z q q q 3 Φ 3 q 6 ; q 3 ; z 3 q 3 q 4 + z1/ (1 ) (1 q ) + z (1 )(1 q ) (1 q )(1 q 3 ) 3 Φ q q 3Φ 3 q 6 ; q 3 ; z 3 q 4 q q 3 q 3 ; q 3 ; z 3 q 5 z 3 < 1 4 Certin trnsformtions nd summtions In this section we shll mke use of (14) nd summtion formule (3) (11) to estblish certin trnsformtion nd summtion formule for q-series If we tke B n 1 z γq/αβ in (14) nd mke use of (3) to sum of inner Φ 1 series we get (41) γ α β; q n (1 γq n ) ( γq/αβ) n q n(n 1)/ q γqα γq/β n (1 γ) γq γq/αβ; q γq/α γq/β; q n q n γq n ; q j A j (wq) j j0 A n (wγq/αβ) n q; q n q α β; q j

9 certin trnsformtion nd summtion formule for q-series 187 Agin tking B n 1 z γ/αβ in (14) nd mking use of (4) in order to sum the inner Φ 1 series we get: (4) γ α β; q n (1 γq n ) ( γq/αβ) n q n(n 1)/ q γq/α γq/β n (1 γ) { } n αβ(1 + γq n ) γq n (α + β) αβ γ A n (wγ/αβ) n q; q n j0 q n γq n ; q j A j (wq) j ) q α β; q j We shll mke use of (41) nd (4) in order to estblish our min results (i) Replcing q by q nd then tking γ b x 4 q α bx q β bx q w 1 nd A j x x q; q j x 4 q ; q j in (41) we get: (43) b x 4 q bx q x q ; q n (1 b x 4 q 4n+ ( ) n q n(n 1)/ q n q bx q 3 b x q ; q n (1 b x 4 q ) q 4 Φ n b x 4 q n+ x x q ; q ; q 3 bx q bx q x 4 q b x 4 q 4 q; q x Φ x q ; q ; q bx q 3 bx q ; q 1 x 4 q Now summing the inner 4 Φ 3 -series on the left hnd side nd Φ 1 on the right hnd side of (43) with the help of (6) nd (3) respectively we get: bq bx (44) Φ q 3 ; q ; x q 1 x q ; q bx q; q x q; q (ii) Tking γ bx q α xq b β xq b A j x xq; q j x q ; q j nd w 1 in (41) we get: (45) bx q xq b xq b; q n (1 bx q n+ ) q n(n 1)/ q n q xq b xq b ; q n (1 bx q ) q n bx q n+ x xq ; q ; q 4 Φ 3 xq b xq b x q bx q 3 q; q xq b xq b; q Φ 1 x xq ; q ; q x q

10 188 remy y denis sn singh sp singh Now summing the inner 4 Φ 3 -series on the left hnd side nd Φ 1 series on the right hnd side of (45) with the help of (7) nd (3) respectively we get the following summtion formul: (46) 3Φ bx q 3 bq xq 3 ; q ; x q 3 xq x q 3 ; q 4 + xq (1 xq) 3 Φ bx q 3 bq bxq 4 ; q ; x q 5 x q 3 bxq ; q 4 bx q 3 xq xq ; q x q xq b xq b; q (iii) Replcing q by q 3 nd then tking γ 3 q 3 α (q) 3/ β (q) 3/ q q ; q 3 j A j nd w 1 in (41) we get: 3/ q 3 3/ q 3 ; q 3 j (47) (48) 3 q 3 (q) 3/ (q) 3/ ; q 3 n (1 3 q 6n+3 ) q 3n q 3n(n 1)/ q 3 (q) 3/ q 3 (q) 3/ q 3 ; q 3 n (1 3 q 3 ) q 5 Φ 3n 3 q 3n+3 q q ; q 3 ; q 3 4 (q) 3/ (q) 3/ 3/ q 3 3/ q 3 3 q 6 q 3 ; q 3 (q) 3/ q 3 (q) 3/ q 3 ; q 3 3Φ q q ; q 3 ; q 3 3/ q 3 3/ q 3 Now summing the inner 5 Φ 4 -series on the right hnd side of (47) with the help of (8) we get: (iv) Tking 1Φ 0 q ; q ; q 3 ; q 3 q 3 3 q 6 ; q 3 (q) 3/ q 3 (q) 3/ q 3 ; q 3 3Φ q q ; q 3 ; q 3 3/ q 3 3/ q 3 (49) γ x 3 q 4 α x 3/ q β x 3/ q w 1 nd A j in (41) we get: x 3 q 4 ; q n q n+n(n 1)/ x ωxq ω xq; q j (xq) 3/ (xq) 3/ ; q j q n x 3 q n+4 x ωxq ω xq ; q ; q x 3/ q x 3/ q (xq) 3/ (xq) 3/ 5Φ 4 q; q n x 3 q 5 q; q x ωxq ω 3Φ xq ; q ; q x 3/ q 3 x 3/ q 3 ; q (xq) 3/ (xq) 3/ where ω e πi/3

11 certin trnsformtion nd summtion formule for q-series 189 Now summing the inner 5 Φ 4 -series on the right hnd side of (49) with the help of (9) we get: x 3Φ 3 q 6 xq 4 xq 5 ; q 3 ; x 3 q 6 xq xq ; q 9 (410) + xq(1 x q 4 ) (1 xq) + x q 3 (1 x q 4 )(1 x q 5 ) (1 xq)(1 xq ) x 3 q 5 q; q x 3/ q 3 x 3/ q 3 ; q 3Φ where ω e πi/3 nd x < 1 q < 1 (v) Lstly tking in (41) we get: 3Φ x 3 q 6 x q 7 xq 5 ; q 3 ; x 3 q 9 xq x q 4 ; q 9 x 3Φ 3 q 6 x q 7 x q 8 ; q 3 ; x 3 q 1 x q 4 x q 5 ; q 9 x ωxq ω xq ; q ; q (xq) 3/ (xq) 3/ γ q α q β q w 1 nd A j 1/3 ω 1/3 ω 1/3 q ; q j q ; q j (411) q; q n q n(n 1)/ q 6Φ n q n+1 1/3 ω 1/3 ω 1/3 q ; q ; q 5 q; q n q q q 1/3 ω 1/3 ω 1/3 q ; q ; q q q q; q q q q q; q 4Φ 3 Now summing the inner 6 Φ 5 -series on the left hnd side of (411) with the help of (11) we get: q 3Φ 3 q q 6 ; q 3 ; q 6 q 3 q 4 ; q 9 + q (1 ) q q q (1 q ) 4 Φ 3 q 6 ; q 3 ; q 9 3 q 4 q 5 ; q 9 (41) + q3 (1 )(1 q ) q q (1 q )(1 q 3 ) 3 Φ 3 q 3 ; q 3 ; q 1 q 5 ; q 9 q q ; q q q q 4Φ 1/3 ω 1/3 ω 1/3 q 3 ; q ; q q; q q where ω e πi/3 < 1 nd q < 1

12 190 remy y denis sn singh sp singh Proceeding in the sme wy one cn lso estblish certin summtion nd trnsformtion formule for q-series by mking use of the summtion (5) (11) nd identity (4) Acknowledgement This work is supported by mjor reserch project No F31-83/005(SR) dted Mrch wrded by the University Grnt Commission Govt of Indi New Delhi for which Remy Denis wishes to express his grtitude to the uthorities concerned The first two uthors re lso thnkful to the Deprtment of Science nd Technology Govt of Indi New Delhi for support under reserch projects No SR/S4/MS-461/07 dtd13008 entitled A study of bsic hypergeometric functions with specil reference to Rmnujn mthemtics nd NoSR/S4/MS:54 dtd10008 entitled Glimpses of Rmnujn s mthemtics in the field of q-series snctioned to them respectively References 1 Biley WN Identities of Rogers-Rmnujn type Proc London Mth Soc (50) (1949) 1 10 Denis RY Certin summtion of q-series nd identities of Rogers Rmnujn type J Mth Phy Sci (1) (1988) Slter LJ Further identities of Rogers-Rmnujn type Proc London Mth Soc (53) (1951) Slter LJ Generlized Hypergeometric Functions Cmbridge University Press Cmbridge Singh UB A note on trnsformtion of Biley Q J Mth Oxford (45) (1994) Singh SP Certin trnsformtion formule for q-series Indin J Pure Appl Mth 31 (10) (000) Verm A On identities of Rogers-Rmnujn type Indin J Pure Appl Mth 11 (6) (1980) Verm A Some trnsformtions of series with rbitrry terms Institute Lombrdo (Rendi Sc) A 106 (197) Verm A nd Jin VK Some summtion formule of bsic hypergeometric series Indin J Pure Appl Mth 11 (8) (1980) Accepted: