ON NEW FORMS OF THE RECIPROCITY THEOREMS

Transcription

1 Gulf Joural of Mathematics Vol 3, Issue ) ON NEW FORMS OF THE RECIPROCITY THEOREMS D. D. SOMASHEKARA 1, K. NARASIMHA MURTHY 2, S. L. SHALINI 3 Abstract. I his lost otebook, Ramauja has stated a beautiful two variable reciprocity theorem. I the recet past, three ad four variable geeralizatios of the reciprocity theorem of Ramauja were established. I this paper we establish ew symmetric ad elegat forms of all the three reciprocity theorems. We also deduce therefrom some q-gamma, q-beta ad eta fuctio idetities. 1. Itroductio ad prelimiaries O page 40 of his lost otebook 14, Ramauja has give the followig beautiful two variable reciprocity theorem. 1 ρa, b) ρb, a) = b 1 ) aq/b, bq/a, q), 1.1) a aq, bq) ρa, b) = ) 1) q +1)/2 a b 1.2) b aq) ad a, b are complex umbers other tha 0 ad q. Throughout this paper, we assume q < 1 ad employ the customary otatios a) := a; q) := 1 aq ), a) := a; q) := a), is a iteger, aq ) a 1, a 2,..., a m ) = a 1 ) a 2 )...a m ), is a iteger or. Adrews 4 was the first to establish 1.1) by employig his four-variable idetity ad the well-kow Jacobi s triple product idetity which is a special case of 1.1). Somashekara ad Fathima 15 used Ramauja s 1 ψ 1 summatio formula ad Heie s trasformatio formula to establish a equivalet versio of 1.1). Bhargava, Somashekara ad Fathima 8 provided aother proof of 1.1). Kim, Somashekara ad Fathima 12 gave a proof of 1.1) usig oly q-biomial theorem. Guruprasad ad Pradeep 10 also have devised a proof of 1.1) usig q-biomial theorem. Adiga ad Aitha 1 established a proof of 1.1) by the method of Date: Received: Sep 4, 2013; Accepted: Feb 24, Correspodig author Mathematics Subject Classificatio. Primary 33D15; Secodary 33D05, 11F20. Key words ad phrases. q-series, reciprocity theorems, q-gamma, q-beta, eta fuctios. 104

2 ON NEW FORMS OF THE RECIPROCITY THEOREMS 105 aalytic cotiuatio. Berdt, Cha, Yeap ad Yee 7 foud three differet proofs of 1.1). Kag 11 costructed a proof of 1.1) alog the lies of Vekatachaliegar s proof of Ramauja s 1 ψ 1 summatio formula. I 17, Somashekara ad Narasimha Murthy gave a proof of 1.1) usig Abel s lemma ad Jacobi s triple product idetity. Recetly, Somashekara, Narasimha Murthy ad Shalii 19 have proved a equivalet form of the geeral idetity of Adrews 4 usig the parameter augumetatio method ad employed the same to derive 1.1). Further, Somashekara ad Narasimha Murthy 18 have give a fiite form of 1.1). For more details oe may refer the book by Adrews ad Berdt 5. Kag 11 has established the followig three ad four variable geeralizatios of 1.1). 1 ρ 3 a, b; c) ρ 3 b, a; c) = b 1 ) c, aq/b, bq/a, q), 1.3) a c/a, c/b, aq, bq) ρ 3 a, b; c) := ) c) 1) q +1)/2 a b 1.4) b aq) c/b) +1 ad c < a < 1 ad c < b < 1. ρ 4 a, b; c, d) ρ 4 b, a; c, d) 1 = b 1 ) c, d, cd/ab, aq/b, bq/a, q), 1.5) a c/a, c/b, d/a, d/b, aq, bq) ρ 4 a, b; c, d) := ad c, d < a, b < ) c, d, cd/ab) 1 + cdq 2 /b) 1) q +1)/2 a b 1.6) b aq) c/b, d/b) +1 To derive 1.3), Kag 11 has employed Ramauja s 1 ψ 1 summatio formula ad Jackso s trasformatio of 2 φ 1 ad 2 φ 2 series. Later, Adiga ad Guruprasad 2 have give a proof of 1.3) usig q-biomial theorem ad q-gauss summatio formula. Somashekara ad Mamta 16 have obtaied 1.3) usig 1.1) by parameter augmetatio method. Oe more proof of 1.3) was give by Zhag 21. Recetly, i 19, Somashekara, Narasimha Murthy ad Shalii have give a proof of 1.3) ad i 18, Somashekara ad Narasimha Murthy have give a fiite form of 1.3). Kag 11 has established the four variable reciprocity theorem 1.5) by employig Adrews geeralizatio of 1 ψ 1 summatio formula 4, Theorem 6, Sear s trasformatio for 3 φ 2 series ad a limitig case of Watso s trasformatio for a termiatig very well-poised 8 φ 7 series. Adiga ad Guruprasad 3 have derived 1.5) usig a idetity of Adrews 4, Theorem 1, Ramauja s 1 ψ 1 summatio formula ad Watso s trasformatio. Recetly, Somashekara,

3 106 D. D. SOMASHEKARA 1, K. NARASIMHA MURTHY 2, S. L. SHALINI 3 Narasimha Murthy ad Shalii 19 have give a proof of 1.5) ad i 18, Somashekara ad Narasimha Murthy have give a fiite form of 1.5). The mai objective of this paper is to give ew symmetric forms of the reciprocity theorems alog with a uified approach to their proofs ad to ivestigate their applicatios. The paper is orgaised as follows. I sectio 2, we give some basic defiitios ad otatios ad state some results which we use i the remaider of the paper. I sectio 3, we state ad prove the symmetric forms of the reciprocity theorems ad i sectio 4, we preset a umber of applicatios of the mai results. 2. Some stadard defiitios ad otatios We recall some basic defiitios ad otatios. q-gamma fuctio 20 is defied by q-beta fuctio 6 is defied by We also have 6 Γ q x) = q) q x ) 1 q) 1 x, 0 < q < ) B q x, y) = 1 q) The Dedekid eta fuctio is defied by ητ) := e πiτ/12 q +1 ) q +y ) q x. 2.2) B q x, y) = Γ qx)γ q y) Γ q x + y). 2.3) =1 1 e 2πiτ ), Imτ) > 0 := q 1/24 q; q), e 2πiτ = q. 2.4) A q-shifted factorial idetity 9, equatioi.17), p.352 is a) +m = a) m aq m ). 2.5) q-biomial theorem 9, equatioii.3), p.354 is give by a) q) z = az) z). 2.6) Heie s trasformatio of 2 φ 1 -series 9, equatioiii.2), p.359 is A, B) q, C) Z = C/B, BZ) C, Z) ABZ/C, B) q, BZ) ) C. 2.7) B

4 ON NEW FORMS OF THE RECIPROCITY THEOREMS 107 Sear s trasformatio of 3 φ 2 series 9, equatioiii.9), p.359 is A, B, C) q, D, E) ) DE ABC = E/A, DE/BC) E, DE/ABC) A, D/B, D/C) q, D, DE/BC) ) E. 2.8) A I 19, Somashekara, Narasimha Murthy ad Shalii have established the followig idetity which is equivalet to the geeral idetity of Adrews 4 5, Theorem6.2.1), p.114. c, d) q c, d) ) bq/d) d = aq, bq) a aq, bq) c/a) +1 a ) c, bq/d) a bq) c/a) Mai results ) d. 2.9) a I this sectio, we establish ew symmetric forms of all the three reciprocity theorems. Theorem 3.1. If a, b < 1 the f 2 a, b) f 2 b, a) = f 2 a, b) = 1 a 1 a 1 ) q, aq/b, bq/a), 3.1) b a, b) ) q b). a Proof. I 2.6), set a = aq m ad shift the ier idex of summatio to + m to obtai aq m ) +m z +m = azq m ). 3.2) q) +m z) = m Usig the q-shifted factorial idetity 2.5) i 3.2), we obtai a) z = az) azq m ) m q) m q m+1 ) z) aq m ) m z m = m = az) q/az) m q) m z) q/a) m. 3.3) Lettig m i 3.3), we obtai a) z = az, q/az, q). 3.4) z, q/a) =

5 108 D. D. SOMASHEKARA 1, K. NARASIMHA MURTHY 2, S. L. SHALINI 3 The equatio 3.4) ca be writte as a) z q 1 1) q +1)/2 q ) az, q/az, q) =. 3.5) az 1 q/a) q 2 /a) az z, q/a) Puttig B = q, C = q 2 /a, Z = q 2 /axz) i 2.7) ad the lettig x, we obtai 1) q +1)/2 q ) = 1 q ) q ) q ). 3.6) q 2 /a) az a z a Usig 3.6) i 3.5), we obtai q ) q ) q ) a) z az) q/az) q) =. 3.7) az z a z) q/a) Replacig a by q/a ad z by b i 3.7), we obtai 3.1), after some simplificatios. This completes the proof. Theorem 3.2. If c < a < 1 ad c < b < 1, the 1 f 3 a, b; c) f 3 b, a; c) = a 1 ) q, aq/b, bq/a, c), 3.8) b a, b, c/a, c/b) f 3 a, b; c) = 1 a q/a) c/a) +1 b). Proof. Settig A = q, B = bq/d, C = c, D = cq/a ad E = bq i 2.8), we obtai bq/d, c) bq) c/a) +1 ) d = 1 + b) a Substitutig 3.9) i 2.9), we obtai c, d) q c, d) = aq, bq) a aq, bq) a ) 1 + b) cd/ab, q/a) c/a, d/a) +1 b). 3.9) bq/d) c/a) +1 d/a) cd/ab, q/a) c/a, d/a) +1 b). 3.10) Iterchagig c ad d i 3.10) ad the settig d = 0 i the resultig idetity, we obtai c) q c) ) ) bq c = aq, bq) a aq, bq) c a ) q/a) 1 + b) b). 3.11) a c/a) +1

6 ON NEW FORMS OF THE RECIPROCITY THEOREMS 109 Iterchagig a ad b i 3.11), we obtai c) q c) = aq, bq) b aq, bq) + ) aq b c ) 1 + a) c Subtractig 3.12) from 3.11), we obtai after some simplificatios 1 a q/a) b) 1 q/b) a) c/a) +1 b c/b) +1 c) 1 ) ) bq c = 1 ) aq a, b) a c a b c b ) q/b) c/b) +1 a). 3.12) ) c. 3.13) b Replacig a by c/b ad b by c/a i 3.1) ad the usig the resultig idetity i 3.13), we obtai 3.8) after some simplificatios. This completes the proof. Theorem 3.3. If c, d < a, b < 1, the f 4 a, b; c, d) f 4 b, a; c, d) 1 = a 1 ) q, aq/b, bq/a, c, d, cd/ab), 3.14) b a, b, c/a, c/b, d/a, d/b) f 4 a, b; c, d) = 1 q/a, cd/ab) b). a c/a, d/a) +1 Proof. Iterchage a ad b i 2.9) to obtai c, d) q = c, d) aq, bq) b aq, bq) b Subtract 3.15) from 3.10) to obtai q/a, cd/ab) b) 1 c/a, d/a) +1 b = c, d) 1 bq/c) a, b) a d/a) +1 1 a ) 1 + a) aq/d) c/b) +1 d/b) q/b, cd/ab) a) c/b, d/b) +1 ) c 1 aq/c) a b d/b) +1 cd/ab, q/b) c/b, d/b) +1 a). 3.15) ) c. 3.16) b Replacig a by c/b ad b by c/a i 3.8) ad usig the resultig idetity i 3.16), we obtai 3.14) after some simplificatios. This completes the proof.

7 110 D. D. SOMASHEKARA 1, K. NARASIMHA MURTHY 2, S. L. SHALINI 3 4. q-gamma, q-beta ad Eta fuctio idetities I this sectio, we deduce some iterestig q-gamma, q-beta ad eta fuctio idetities usig symmetric forms of the reciprocity theorems. Equatio 3.1) ca be writte as ) q a) b ) q b) = q, aq/b, b/a). 4.1) b a a a, b) Settig a = q x ad b = q x+y i 4.1), we obtai ) q 1 x y qx q ) y q 1 x qx+y ) = q, q1 y, q y ). 4.2) q x, q x+y ) Usig 2.1) i 4.2), we obtai after some simplificatios Γ q x) Γ q 1 y)γ q y) = qx+y ) ) q 1 x y 1 q) x q) 2 qx q ) y q 1 x qx+y ). Similarly, settig a = q x ad b = q 3x i 4.1) ad the usig 2.1) i the resultig idetity, we obtai after some simplificatios Γ q x)γ q 3x) 1 q)1 4x = ) q 1 3x Γ q 2x)Γ q 1 2x) q) qx q ) 2x q 1 x q3x. Settig a = q x ad b = q 1 y i 4.1) ad the usig 2.1), we obtai after some simplificatios Γ q x)γ q 1 y) = 1 q)1+y x q) q y+x, q 1 y x ) q y ) q x q ) 1 y x q 1 x q1 y ). Chagig q to q 2 ad the settig a = q ad b = q 2 i 4.1) ad the usig 2.4), we obtai after some simplificatios η 7 2τ) η 3 τ)η 3 4τ) = q 1/24 2 q 2 ; q 2) 1 q + 1) q; q 2 ) q 2+1. Chagig q to q 2 ad the settig a = q ad b = q i 4.1) ad the usig 2.4), we obtai after some simplificatios η 3 4τ) η 2 2τ) = q1/3 1) q; q 2 ) q + ) q; q 2 2 q. Settig a = q 1/2 ad b = q ad the chagig q to q 2 i 4.1) ad the usig 2.4), we obtai after some simplificatios η 3 τ) η 2 2τ) = q 1/24 2 1) q 2 ; q 2 ) 1 q 1) q; q 2 ) q 2+1.

8 ON NEW FORMS OF THE RECIPROCITY THEOREMS 111 Chagig q to q 2 ad the settig a = q ad b = q 2 i 4.1), we obtai after some simplificatios χq) = ) q; q 2 q2+1, χq) = q; q 2 ), which ca be foud i Ramauja s otebook 13, p.197. Settig a = q 1/3 ad b = q 2/3 ad the chagig q to q 3 i 4.1), we obtai after some simplificatios η 2 τ)η6τ) η2τ)η3τ) = q1/8 1) q; q 3 ) q 1) q 2 ; q 3 ) q 2+1. Equatio 3.8) ca be writte as q/b) a) b c/b) +1 a q/a) c/a) +1 b) = q, aq/b, b/a, c) a, b, c/a, c/b). 4.3) Settig a = q x, b = q y ad c = q x+y i 4.3), we obtai q 1 y ) q x q y x q 1 x ) q y = q, q1+x y, q y x, q x+y ). 4.4) q x ) +1 q y ) +1 q x, q y ) 2 Usig 2.1) ad 2.3) i 4.4), we obtai after some simplificatios Bq 2 x, y) = 1 q)2 q) q x+y ) q 1 y ) q x q y x q 1 x ) q y. q 1+x y, q y x ) q x ) +1 q y ) +1 Similarly, settig a = q x, b = q y, c = q 2x i 4.3), usig 2.1) ad 2.3), we obtai after some simplificatios B q x, y) = 1 q)qx, q 2x y, q x+y ) q 1+x y, q y x, q 2x ) q 1 y ) q x q y x q 2x y ) +1 q 1 x ) q x ) +1 q y Settig a = q x, b = q 1 x, c = q 2x i 4.3) ad the usig 2.1), we obtai after some simplificatios Γ 2 qx)γ q 1 x)γ q 3x 1) Γ 2 q2x)γ q 1 2x) = 1 q) 2 2x q x ) q 3x 1 ) +1 q x q 1 2x q 1 x ) q x ) +1 q 1 x )..

9 112 D. D. SOMASHEKARA 1, K. NARASIMHA MURTHY 2, S. L. SHALINI 3 Settig a = q x, b = q y, c = q 2x+y i 4.3) ad the usig 2.1), we obtai after some simplificatios Γ q x)γ q 2x)Γ q y) = 1 q)3 3x y q) 2 q x+y ) q 1+x y, q y x, q 2x+y ) q 1 y ) q x q y x q 1 x ) q y. q 2x ) +1 q x+y ) +1 Settig a = q x, b = q 2x+y, c = q 3x i 4.3) ad the usig 2.1), we obtai after some simplificatios Γ q x)γ q 2x) Γ q 3x) = 1 q)q2x+y, q x y ) q 1 x y, q x+y ) q 1 2x y ) q x q x+y q x y ) +1 q 1 x ) q 2x ) +1 q 2x+y ) Settig a = q x, b = q x+y, c = q 2x i 4.3) ad the usig 2.1), we obtai after some simplificatios Γ 2 qx) Γ q 2x)Γ q y)γ q 1 y) = qx+y, q x y ) q) 2 q 1 x y ) q x q y q x y ) +1 q 1 x ) q x ) +1 q x+y ) Settig a = q x, b = q 2x, c = q 3x i 4.3) ad the usig 2.1), we obtai after some simplificatios Γ q x)γ 2 q2x) q 1 2x ) = 1 q)2 3x q x q x q 1 x ) q 2x. Γ q 3x)Γ q 1 x) q x ) +1 q 2x ) +1 Settig a = q 1 x, b = q 2x 1, c = q 2x i 4.3) ad the usig 2.1), we obtai after some simplificatios Γ q 1 x) Γ q 3 3x) = 1 q2x 1 )1 q) 2 2x 1 q 3x 2 ) q 2 2x ) q 1 x ) q 3x 2 q) +1 q x ) q 3x 1 ) +1 q 2x 1 ) Settig a = q x, b = q 1 y, c = q 1 x i 4.3) ad the usig 2.1), we obtai after some simplificatios Γ q x)γ q 1 y)γ q 1 2x) = 1 q)1+y q) q y x ) Γ q 1 x) q x+y, q 1 y x ) q y ) q x q 1 x y q 1 x ) q 1 y ). q y x ) +1 q 1 2x ) +1...

10 ON NEW FORMS OF THE RECIPROCITY THEOREMS 113 Chagig q to q 2 ad the settig a = q, b = c = q 2 i 4.3) ad the usig 2.4), we obtai after some simplificatios φ 2 q q) = q 2 ) + 1) q 2, 1 q 2+1 ) φq) = = q 2 = q; q2 ) q 2 ; q 2 ) q; q 2 ) q 2 ; q 2 ) is oe of the theta fuctios i Ramauja s otebook 13, p.197. Chagig q to q 2 ad the settig a = q, b = c = q 2 i 4.3) ad the usig 2.4), we obtai after some simplificatios η 3 2τ) η 2 τ)η4τ) = 1 2 q; q 2 ) q; q 2 ) +1 q 2+1. Chagig q to q 2 ad the settig a = q, b = q ad c = q 2 i 4.3) ad the usig 2.4), we obtai after some simplificatios η 4 4τ) η 3 2τ) = q5/12 q; q 2 ) 1) q q; q 2 ) + q. 2 q; q 2 ) +1 q; q 2 ) +1 Chagig q to q 2 ad the settig a = q, b = q ad c = q 3 i 4.3), we obtai after some simplificatios η 2 4τ)ητ) = q1/8 q; q 2 ) 1) q q; q 2 ) + q. η 3 2τ) 2 q 2 ; q 2 ) +1 q 2 ; q 2 ) +1 Chagig q to q 2 ad the settig a = q 2, b = q 2 ad c = q 3 i 4.3) ad the usig 2.4), we obtai after some simplificatios η4τ) ητ) = q 2 ; q 2 ) 1 q1/8 1 + q) q 2. q; q 2 ) +1 Equatio 3.14) ca be writte as q/b, cd/ab) c/b, d/b) +1 a) b a = q/a, cd/ab) c/a, d/a) +1 b) q, aq/b, b/a, c, d, cd/ab) a, b, c/a, c/b, d/a, d/b). 4.5) Settig a = q 2x, b = q 2y, c = q 3x ad d = q 3y i 4.5), we obtai q 1 2y, q x+y ) q 3x 2y, q y ) +1 q 2x q 2y 2x q 1 2x, q x+y ) q x, q 3y 2x ) +1 q 2y = q, q1+2x 2y, q 2y 2x, q 3x, q 3y, q x+y ) q 2x, q 2y, q x, q y, q 3x 2y, q 3y 2x ). 4.6)

11 114 D. D. SOMASHEKARA 1, K. NARASIMHA MURTHY 2, S. L. SHALINI 3 Usig 2.1) ad 2.3) i 4.6), we obtai after some simplificatios B q x, y)b q 2x, 2y) = 1 q)q3x 2y, q 3y 2x, q 2x+2y ) B q 3x, 3y) q 1+2x 2y, q 2y 2x, q 3x+3y ) q 1 2y, q x+y ) q 2x q 2y 2x q 1 2x, q x+y ) q 2y. q 3x 2y, q y ) +1 q x, q 3y 2x ) +1 Similarly, settig a = q x, b = q y, c = d = q 2x i 4.5) ad the usig 2.1) ad 2.3), we obtai after some simplificatios B q x, y) = 1 q)qx+y ) q 2x y, q x ) 2 q 1+x y, q y x, q 3x y ) q 2x ) 2 q 1 y, q 3x y ) q x q y x q 2x y ) 2 +1 q 1 x, q 3x y ) q y. q x ) 2 +1 Settig a = q x, b = q 1 y+x, c = d = q x+y i 4.5) ad the usig 2.1) ad 2.3), we obtai after some simplificatios Bq 2 x, y) B q x, 1 y) = 1 q)q2y 1 ) 2 q y, q 3y 1 ) q y x, q 3y 1 ) q x q 1 y q 2y 1 ) 2 +1 q 1 x, q 3y 1 ) q 1 y+x ). q y ) 2 +1 Settig a = q x, b = q 1 x, c = d = q 1+y i 4.5) ad the usig 2.1), we obtai after some simplificatios Γ 3 qx)γ q 1 x) Bq 2 x, y)γ q 2x)Γ q 1 2x)Γ q 1 + 2y) = 1 q1+y x ) 2 1 q y ) 2 q x, q 1+2y ) q x q 1 2x q 1 x, q 1+2y ) q 1 x ). q x+y ) 2 +1 q 1+y x ) 2 +1 Settig a = q x 1, b = q 2x 1, c = d = q 2x i 4.5) ad the usig 2.1), we obtai after some simplificatios Γ q x + 1) Γ q 1 x)γ q 2x) = 1 qx 1 )1 q 2x 1 )1 q x+1 ) 1 q)q) q 2 2x, q x+2 ) q x 1 ) q x q 2 x, q x+2 ) q 2x 1 ). q) 2 +1 q x+1 ) 2 +1

12 ON NEW FORMS OF THE RECIPROCITY THEOREMS 115 Settig a = q 1 2x, b = q 1 x, c = d = q x i 4.5) ad the usig 2.1), we obtai after some simplificatios Γ q 1 2x)Γ 2 q3x 1)Γ 2 q2x 1) Γ 3 qx)γ q 5x 2) = 1 q) 2 q x, q 5x 2 ) q 1 2x ) q x q 2x 1 ) 2 +1 q 2x, q 5x 2 ) q 1 x ). q 3x 1 ) 2 +1 Settig a = q 1/2, b = q 1/2, c = d = q ad the chagig q to q 2 i 4.5) ad the usig 2.4), we obtai after some simplificatios η 8 4τ) η 7 2τ) = q3/4 q, q 2 ; q 2 ) q 1) q, q 2 ; q 2 ) + q. 2 q; q 2 ) 2 +1 q; q 2 ) 2 +1 Settig a = q 1/2, b = q 1/2, c = d = q 3/2 ad the chagig q to q 2 i 4.5) ad the usig 2.4), we obtai after some simplificatios 1 1 q)2 q; q 2 ) = q q; q 2 ) q 2 ; q 2 ) +1 1) + q. φq) 2 q 2 ; q 2 ) +1 q 2 ; q 2 ) 2 +1 Chagig q to q 2 ad the settig a = q 2, b = q 2, c = d = q 3 i 4.5) ad the usig 2.4), we obtai after some simplificatios η 2 τ)η 4 4τ) η 6 2τ) = q 1/4 1 q) 2 q 2 ; q 2 ) 1 q 2 ; q 2 ) q 2. q; q 2 ) 2 +1 Chagig q to q 2 ad the settig a = q, b = q 2, c = d = q 3 i 4.5) ad the usig 2.4), we obtai after some simplificatios η 2 2τ) η 2 τ)η 2 4τ) = q 1/4 1 + q) 2 q; q 2 ) q; q 2 ) +1 q 2+1. q 2 ; q 2 ) 2 +1 Chagig q to q 2 ad the settig a = 1, b = q, c = d = q 2 i 4.5) ad the usig 2.4), we obtai after some simplificatios η 6 τ) η 3 2τ) = q3/4 q 2 ; q 2 ) q q 2 ; q 2 ) 1) + q q 2+1 ) q; q 2 ) +1 1 q 2+1 )q; q 2 ) +1 Ackowledgemet. The first author is thakful to Uiversity Grats CommissioUGC), Idia for the fiacial support uder the grat SAP-DRS-1-NO.F.510/2/DRS/2011. The secod author is thakful to Uiversity Grats Commissio, Idia, for the award of Teacher Fellowship uder the grat No.KAMY074-TF ad the third author is thakful to UGC for awardig the Rajiv Gadhi Natioal Fellowship, No.F1-17.1/ /RGNF-SC- KAR-2983/SA-III/Website).

13 116 D. D. SOMASHEKARA 1, K. NARASIMHA MURTHY 2, S. L. SHALINI 3 Refereces 1. C. Adiga ad N. Aitha, O a reciprocity theorem of Ramauja, Tamsui J. Math. Sci ), C. Adiga ad P. S. Guruprasad, O a three variable reciprocity theorem, South East Asia Joural of Mathematics ad Mathematical Scieces 62)2008), C. Adiga ad P. S. Guruprasad, A ote o four variable reciprocity theorem, It. Jour. Math. Math Sci. Vol doi: 10, 1155/2009/ G. E. Adrews, Ramauja s lost otebook. I. partial θ-fuctios, Adv. i Math ), G. E. Adrews ad B. C. Berdt, Ramauja s Lost Notebook, Part II, Spriger, New York, R. Askey, The q-gamma ad q-beta fuctios, Applicable Aal ), B. C. Berdt, S. H. Cha, B. P. Yeap ad A. J. Yee, A reciprocity theorem for certai q-series foud i Ramauja s lost otebook, Ramauja J ), S. Bhargava, D. D. Somashekara ad S. N. Fathima, Some q-gamma ad q-beta fuctio idetities deducible from the reciprocity theorem of Ramauja, Adv. Stud. Cotemp. Math. Kyugshag) ), G. Gasper ad M. Rahma, Basic hypergeometric series, Ecyclopedia of Mathematics, Cambridge Uiversity press. Cambridge, P. S. Guruprasad ad N. Pradeep, A simple proof of Ramauja s reciprocity theorem, Proc. Jagjeo Math. Soc ), S.-Y. Kag, Geeralizatios of Ramauja s reciprocity theorem ad their applicatios, J. Lodo Math. Soc ), T. Kim, D. D. Somashekara ad S. N. Fathima, O a geeralizatio of Jacobi s triple product idetity ad its applicatio, Adv. Stud. Cotemp. Math. Kyugshag) 92004), S. Ramauja, Notebooks 2 Volumes), Tata Istitute of Fudametal Research, Bombay, S. Ramauja, The lost otebook ad other upublished papers, Narosa, New Delhi, D. D. Somashekara ad S. N. Fathima, A iterestig geeralizatio of Jacobi s triple product idetity, Far East J. Math. Sci ), D. D. Somashekara ad D. Mamta, O the three variable reciprocity theorem ad its applicatios, Aus. jour. Math. aal. appl. 91)Art.13)2012), D. D. Somashekara ad K. Narasimha murthy, O A Two Variable reciprocity theorem Of Ramauja Bull. Pure Appl. Math. 51)2011), D. D. Somashekara ad K. Narasimha murthy, Fiite forms of reciprocity theorems, Iteratioal J.Math. Combi ), D. D. Somashekara, K. Narasimha murthy ad S. L. Shalii, O the reciprocity theorem of Ramauja ad its geeralizatios, Proc. Jagjeo Math. Soc. 153)2012), J. Thomae, Beiträge zur theorie der durch die Heiesche reie..., J. Reie Agew. Math ), Z.-Z. Zhag, A ote o a idetity of Adrews with Erratum), Electro. J. Combi ), N3.

14 ON NEW FORMS OF THE RECIPROCITY THEOREMS Departmet of Mathematics, Uiversity of Mysore,Maasagagotri, Mysore , INDIA. address: dsomashekara@yahoo.com 2 Departmet of Mathematics, Uiversity of Mysore,Maasagagotri, Mysore , INDIA. address: simhamurth@yahoo.com 3 Departmet of Mathematics, Uiversity of Mysore,Maasagagotri, Mysore , INDIA. address: shaliisl.maths@gmail.com