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CERTAIN MODULAR RELATIONS FOR REMARKABLE PRODUCT OF THETA-FUNCTIONS
M. S. MAHADEVA NAIKA,K. Sushan Bairy,N. P. SUMAN 장전수학회 2014 Proceedings of the Jangjeon mathematical society Vol.17 No.3
At scattered places of his second notebook, Ramanujan recorded several P Q mixed modular equations with four moduli. In this paper, we establish several new P Q mixed modular equations analogous to those recorded by Ramanujan in his notebooks. Employing these, we establish new modular relations for Ramanujan's remarkable product of theta-functions.
New identities for ratios of Ramanujan’s theta function
M. S. MAHADEVA NAIKA,S. Chandankumar,K. Sushan Bairy 장전수학회 2017 Advanced Studies in Contemporary Mathematics Vol.27 No.1
Ramanujan in his notebooks, has established several new modular equation which he denoted as P and Q. In this paper, we establish several new identities for ratios of Ramanujan’s theta function involving φ(q). We establish some new explicit evaluations for the ratios of Ramanujan’s theta function. We also establish some new modular relations for a continued fraction of order twelve H(q) with H(qn) for n =2, 4, 6, 8, 10, 12, 14 and 16.
Certain quotient of eta-function identities
M.S. Mahadeva Naika,M.C. Maheshkumar,K. Sushan Bairy 장전수학회 2008 Advanced Studies in Contemporary Mathematics Vol.16 No.1
On page 212 in his lost notebok, Ramanujan defined a parameter λn by a certain quotient of Dedekind eta-functions at the argument q = exp(πpn/3). He then recorded a table of several values of λn := λn, 3. All these have been established by B. C. Berndt, H. Chan, S.-Y. Kang and L.-C. Zhang [4].λn,p at the argument q = exp(πpn/p ). We establish several interesting and new explicit evaluations for λn, p using Ramanujan-Weber class invariant,modular equations and mixed-modular equations.