A NEW CLASS OF MODULAR EQUATIONS IN RAMANUJAN S ALTERNATIVE THEORY OF ELLIPTIC FUNCTIONS OF SIGNATURE 4 AND SOME NEW P-Q ETA-FUNCTION IDENTITIES

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1 A NEW CLASS OF MODULAR EQUATIONS IN RAMANUJAN S ALTERNATIVE THEORY OF ELLIPTIC FUNCTIONS OF SIGNATURE AND SOME NEW P-Q ETA-FUNCTION IDENTITIES S. Bhagava Chasheka Adiga M. S. Mahadeva Naika. Depatent of Studies in Matheatics Univesity of Mysoe Manasa Gangothi Mysoe (INDIA Abstact: In this pape we obtain a class of odula equations in Raanujan s altenative theoy of elliptic functions of signatue eploy the to obtain a new class of P-Q eta-function identities with fou oduli akin to Raanujan s. Key wods: Elliptic functions odula equations P-Q eta-functions. 000 AMS Matheatics Subject Classification: S D5 D0.. Intoduction In his faous pape Modula Equations Appoxiation to π [6] [ pp.-9] on pages 57-6 of his Second Notebook [7] Raanujan gives an outline of theoies of elliptic functions to altenative bases coesponding to the classical theoy by way of stateents of soe esults. Poofs of all these esults can be found in one of B. C. Bendt s books [ pp ]. Soe of the esults in altenative theoies wee also peviously exained by K. Venkatachalienga [9 pp.9-95] J. M. Bowein P. B. Bowien [5]. In Section of the pesent pape we establish a class of odula equations belonging to altenative theoy of signatue. These equations copleent the known classes of odula equations [7 pp.57 6] [ pp.9-6].

2 In enties 5 7 of Chapte 5 of his Second Notebook [7] Raanujan states twenty thee elegant so called P Q eta-function identities. Eleentay poofs of eighteen of these P Q identities eploying vaious odula equations of Raanujan have been given fo the fist tie in the woks of Bendt [ pp.0-7] Bendt L.-C. Zhang []. In Section we obtain a new class of P-Q identities on eploying the odula equations of theoy of signatue established in Section.. Soe odula equations in the theoy of signatue Let x F x Z Z ; ; : ; ( : ( ;; ;; : ( : x F x F csc exp x q q π π whee 6 0 < x <. Let n denote a fixed natual nube assue that ; ;; ;; ;; F F F F n (. whee o 6. Then a odula equation of degee n in theoy of elliptic functions of signatue is a elation between induced by (.. We often say that is of degee n ove ; ( ; ( : ( Z Z is called the ultiplie. We also use the notations ; ( : : Z Z Z Z n : Z n ( : Z ( ; to indicate that has degee n ove. When the context is clea we oit the aguent in q Z( (.

3 We now collect in the following theoe soe of Raanujan s odula equations belonging to the theoy of signatue (classical theoy. Theoe.. The following odula equations hold in the theoy of signatue (classical theoy. If ae of thid eleventh thity-thid degees ove espectively then (i. ( ( ( ( ( ( ( ( ( ( ( ( (. (ii. ( ( ( ( ( ( ( ( ( ( ( ( (. whee ae ultiplies associated with the pais espectively. If ae of thid ninth degees ove espectively then (iii ( ( ( ( ( ( (. ( ( ( ( (iv ( ( (.5 whee ae ultiplies associated with the pais espectively.

4 If ae of thid fifth fifteenth degees ove espectively then ( ( ( ( (v ( ( ( ( ( ( ( ( (vi ( ( ( ( (.6 (.7 (vii (viii ( ( ( ( ( ( ( ( 6 ( ( ( ( ( ( ( ( ( ( ( ( (. 6 ( ( 9 (.9 ( ( whee ae ultiplies associated with the pais espectively. If ae of thid seventh twentyfist degees ove espectively then (ix ( ( ( ( ( ( ( ( ( ( ( ( 6 (.0 (x ( ( ( ( ( ( ( ( ( ( ( ( 6 (.

5 (xi ( ( ( ( ( ( ( ( ( ( ( ( ZZ Z 7Z (. (xii ( ( ( ( ( ( ( ( ( ( ( ( Z 7Z 7 ZZ (. whee ae ultiplies associated with the pais espectively. If ae of fifth seventh thity-fifth degees ove espectively then ( ( ( ( (xiii ( ( ( ( ( ( ( ( (. (xiv ( ( ( ( ( ( ( ( ( ( ( ( (.5 whee ae ultiplies associated with the pais espectively. 5

6 If ae of thid thiteenth thity-nineth degees ove espectively then (xv ( ( ( ( ( ( ( ( ( ( ( ( (.6 (xvi ( ( ( ( ( ( ( ( ( ( ( ( (.7 whee ae ultiplies associated with the pais espectively. Poof. Fo poofs of (i (ii see [ Enty (i (ii p.0] ; fo poofs of (iii (iv see [ Enty (xii (xiii pp.5-5] : fo poofs of (v (vi (vii (viii see [ Enty (viii (ix (x (xi p.] ; fo poofs of (ix (x (xi (xii see [ Enty (i (ii (iii (iv p.0] ; fo poofs of (xiii (xiv see [ Enty (v (vi p.] ; lastly fo poofs of (xv (xvi see [ Enty 9 (iv p.6]. While outlining his theoy of signatue Raanujan indicates a device of deducing foulas in the theoy of signatue fo coesponding foulas in the classical theoy. In fact if we eplace then x x by x (. Z( ; x gets eplaced by x Z( ; x (.9 q ( x gets eplaced by q (x (.0 6

7 any foula Ω ( x q Z( 0 in the classical theoy yields the foula Ω x q x Z( x 0 in the theoy of signatue. One ay see [ pp. 5-6] fo details whee Bendt also establishes vaious Raanujan s odula equations in the theoy of signatue by this echanis. The odula equations in the theoy of signatue stated in the Theoe. below ae obtained in exactly the sae fashion stating fo the classical odula equations of Theoe.. Theoe.. The following odula equations hold in the theoy of signatue. If ae of thid eleventh thitythid degees ove espectively then (i ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (. (ii ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (. whee ae ultiplies associated with the pais espectively. 7

8 If ae of thid ninth degees ove espectively then (iii ( ( ( ( ( ( ( ( ( (. (iv ( ( ( ( ( ( ( ( ( (. whee ae ultiplies associated with the pais espectively. If ae of thid fifth fifteenth degees ove espectively then (v ( ( ( ( ( ( ( ( ( ( ( ( (.5 (vi ( ( ( ( ( ( ( ( ( ( ( ( (.6

9 9 (vii ( ( ( ( ( ( ( ( ' ( ( ( ( ( ( ( ( 6 (.7 (viii ( ( ( ( ( ( ( ( 9 ( ( ( ( ( ( ( ( 6 (. whee ae ultiplies associated with the pais espectively. If ae of thid seventh twentyfist degees ove espectively then (ix ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( 6 (.9 (x ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( 6 (.0 (xi ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (.

10 (xii ( ( ( ( ( ( ( ( ( ( ( ( 7 ( ( ( ( (. whee ae ultiplies associated with the pais espectively. If ae of fifth seventh thity-fifth degees ove espectively then (xiii ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (. (xiv ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (. whee ae ultiplies associated with the pais espectively. If ae of thid thiteenth thity ninth degees ove espectively then (xv ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (.5 0

11 (xvi ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (.6 whee ae ultiplies associated with the pais espectively. Poof. In view of the eaks ade iediately peceeding the stateent of the theoe we deonstate the poof of only two of the identities (. (.6. Fo exaple to pove (. effect tansfoations (. (.9 (.0 epeatedly on (. of Theoe.. We then obtain ( '( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (. (.7 ( ( Multiplying both sides of (.7 by we obtain ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (. ( ( This copletes the poof of (..

12 To pove (. effect tansfoations (. (.9 (.0 epeatedly on (.5 of Theo.. We then obtain ( ( ( ( ( ( ( ( ( ( ( ( ( (. (. ( ( ( Multiplying both sides of (. by ( we obtain ( ( ( ( ( ( ( ( ( This copletes the poof of (.. (. (. Soe P-Q eta-function identities with fou oduli In this section we obtain algebaic identities between functions of the type. p n z nz z nz ( z : o z nz η ( z η ( nz Q n ( z : P n (z. Hee n ae cetain positive integes z sts as usual fo the Dedekind s eta-function defined by πiz η ( z : q f ( q q e I z > 0 (. whee f is the theta function in Raanujan s notations given by f ( q : Π( q n q <. (.

13 Ou poof consists in eleentay algebaic anipulations of the odula equations in the theoy of signatue stated in Theoe.. We will obtain identities (.-(.0 of which the fist six appea new to liteatue the last two (.9 (.0 ae due to Raanujan poofs eploying Raanujan s odula equations in the theoy of signatue can be found in Bundt s book [ pp.-] whee he also establishes seveal othe Raanujan s P-Q identities with two oduli. Theoe.. If f(-q z ae as defined in (. (. then the following P-Q identities hold with Q (q : P (q always. (i P Q PQ Q P ( P Q PQ whee z z f ( q f ( q P : z z qf ( q f ( q (. P Q P Q (ii PQ PQ Q P Q P whee z 7z P : (. z z (iii P Q 7PQ Q P ( P Q PQ whee z z P : (.5 7z z (iv P Q PQ Q P ( P Q PQ whee z z P : (.6 9z z (v P Q PQ Q P ( P Q PQ whee 7z z P : (.7 5z 5z

14 Q P (vi PQ PQ P Q whee z z P : (. z 9z Q P (vii PQ PQ P Q whee 5z z P : (.9 5z z 9 Q P Q P (viii PQ PQ P Q P Q whee z 5z P :. (.0 z 5z Poof. We epeatedly use the following elations of the theoy of signatue naely q f ( q Z x ( x (. q f ( q Z x ( x (. stated by Raanujan [7 p.60] poved in Bendt s book [ p.] whee x Z: Z ( q q ae elated as in Section. A geneal featue of ou poof is that each of (. (.0 is deived fo a suitable pai of identities taken fo (. (.6. Since the deivations ae siila we deonstate just two cases. Fo instance to pove (. we begin by ewiting P Q of (. as follows by epeated use of (. (.: ( ( P : (. ( (

15 ( ( Q : Q( q : P( q ( (. (. Fo (. (. we deduce that P Q ( ( ( ( (.5 Q P. (.6 Multiplying (. by obtain ( ( ( ( adding the esulting identity to (. we ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (. (.7 Eploying (.5 (.6 in (.7 we obtain the algebaic elation (. between P Q. To pove (. ultiply (. by (. to obtain ( ( ( add the esulting identity to ( ( ( (. ( ' ( We ewite P Q of (. as follows by epeated use of (. (.: (. : ( ( ( P (.9 5

16 ( ( Q : Q( q : P( q (. (.0 Fo (.9 (.0 we deduce that P Q ' ( ( ( (. Q. (. P ' Eploying (. (. in (. we obtain the algebaic elation (. between P Q. Refeences. B. C. Bendt Raanujan s Notebooks Pat III Spinge-Velag New Yok 99.. B. C. Bendt Raanujan s Notebooks Pat IV Spinge-Velag New Yok 99.. B. C. Bendt Raanujan s Notebooks Pat V Spinge-Velag New Yok 99.. B. C. Bendt L.-C. Zhang Raanujan s identities fo eta-functions Math. Ann. 9 ( J. M. Bowein P. B. Bowien A cubic countepat of Jacobi s identity the AGM Tans. Ae. Math. Soc. ( S. Raanujan Modula equations appoxiation to π Quat J. Math. (Oxfod 5 ( S. Raanujan Notebooks ( volues Tata Institute of Fundaental Reseach Bobay S. Raanujan Collected Papes Chelsea New Yok K. Venkatachalienga Developent of Elliptic Functions Accoding to Raanujan Technical Repot Maduai Kaaaj Univesity Maduai 9. 6

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