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General Formulas for Explicit Evaluations of Ramanujan's Cubic Continued Fraction
Naika, Megadahalli Sidda Naika Mahadeva,Maheshkumar, Mugur Chinna Swamy,Bairy, Kurady Sushan Department of Mathematics 2009 Kyungpook mathematical journal Vol.49 No.3
On page 366 of his lost notebook [15], Ramanujan recorded a cubic continued fraction and several theorems analogous to Rogers-Ramanujan's continued fractions. In this paper, we derive several general formulas for explicit evaluations of Ramanujan's cubic continued fraction, several reciprocity theorems, two formulas connecting V (q) and V ($q^3$) and also establish some explicit evaluations using the values of remarkable product of theta-function.
On some new modular equations of degree 9 and their applications
M. S. M. Naika,S. Chandankumar,M. Manjunatha 장전수학회 2012 Advanced Studies in Contemporary Mathematics Vol.22 No.1
In this paper, we establish several new modular equations of degree 9 using Ramanu-jan's modular equations. We also establish several new general formulas to compute the values for r9;n and r09;n. As an application, we establish explicit evaluations of Ramanujan's remarkable product of theta{functions.
ON SOME NEW MODULAR RELATIONS FOR RAMANUJAN'S REMARKABLE PRODUCT OF THETA FUNCTIONS
M. S. MAHADEVA NAIKA,S. Chandankumar,B. HEMANTHKUMAR 장전수학회 2013 Advanced Studies in Contemporary Mathematics Vol.23 No.3
In this paper, we establish several new P{Q mixed modular equations involving theta{ functions which are similar to those recorded by Ramanujan in his notebooks. As an application, we establish several new general formulas for explicit evaluations of Ramanujan's remarkable product of theta{functions and a remarkable product of theta{functions.
SOME NEW EXPLICIT VALUES FOR RAMANUJAN CLASS INVARIANTS
M. S. MAHADEVA NAIKA 장전수학회 2010 Advanced Studies in Contemporary Mathematics Vol.20 No.4
At scattered places in his first notebook, Ramanujan recorded the values for 107 class invariants or irreducible monic polynomials satisfied by them. On pages 294-299 in his second notebook, Ramanujan gave a table of values for 77 class invariants. The main purpose of this paper is to establish some new values for Ramanujan's class invariants using modular equations of degrees 19, 23 and 59.
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.
On some new mixed modular equations of composite degrees 45 and 63
M. S. MAHADEVA NAIKA,M. Prasad 장전수학회 2023 Proceedings of the Jangjeon mathematical society Vol.26 No.2
On some new mixed modular equations of composite degrees 45 and 63
Certain identities for a continued fraction of Ramanujan
M. S. M. Naika,K. S. Bairy,S. Chandankumar 장전수학회 2014 Advanced Studies in Contemporary Mathematics Vol.24 No.1
Ramanujan has recorded several continued fractions in his notebooks. In this paer, we establish several identities of a continued fraction of Ramanujan V(q). We also establish several relations between V(q) and V(qn) and several explicit evaluations of V(q).
On some new Schläfli type mixed modular equations
M. S. M. Naika,K. S. Bairy 장전수학회 2011 Advanced Studies in Contemporary Mathematics Vol.21 No.2
On pages 86 and 88 of his rst notebook, Ramanujan recorded eleven Schla i-type modular equations for composite degrees. Out of eleven, ten have been proved by Berndt [5] using theory of modular forms. In this paper, we establish several new Schla i-type mixed modular equations.
Explicit evaluations of Ramanujan's remarkable product of theta-function
M. S. M. Naika,M. C. Maheshkumar 장전수학회 2006 Advanced Studies in Contemporary Mathematics Vol.13 No.2
On pages 338 and 339 in his first notebook [9], Ramanujan records eighteen values for a certain product of theta-function. All these have been proved by B. C. Berndt,H. Chan and L-C. Zhang [7]. The main purpose of this paper is to establish several new explicit evaluations of Ramanujan’s remarkable product of theta-function.
On 5-regular bipartitions with odd parts distinct
M. S. MAHADEVA NAIKA,HARISHKUMAR T. 장전수학회 2019 Proceedings of the Jangjeon mathematical society Vol.22 No.2
In his work, K. Alladi [1] considered the partition function pod(n), the number of partitions of an integer n with odd parts distinct (the even parts are unrestricted). Later Hirschhorn and Sellers [8] obtained some internal congruences involving innite families of Ramanujan-type congruences for pod(n). Let Bo(n) denote the number of 5-regular bipartitions of a positive integer n with odd parts distinct. In this paper, we establish many innite families of congruences modulo powers of 2 for Bo(n). For example, Bo 32 34 52+2 72 n + t7 34 52+1 72 1 0 (mod 16); for ; ; 0 and t7 2 f28; 92; 124; 156g.