Some modular equations analogous to Ramanujan’s identities

Srivástava, H. M.; Kumar, B. R. Srivatsa; Narendra, R.

doi:10.1007/s13398-021-01002-w

Cited by 3 publications

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“…In conclusion, as per our observations, the general contiguous extensions for the three q-Kummer summation theorems, which are presented in this paper in the form of Theorems 1 to 6 have not appeared previously in the literature. For some recent developments on the basic (or q-) analogues of the known results on summations, transformations and reductions for ordinary hypergeometric series in one, two and more variables (see, especially for a variety of techniques used in the q-analysis [21][22][23][24][25]).…”

Section: Introduction Motivation and Preliminariesmentioning

confidence: 99%

General Summation Formulas Contiguous to the q-Kummer Summation Theorems and Their Applications

et al. 2021

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This paper provides three classes of q-summation formulas in the form of general contiguous extensions of the first q-Kummer summation theorem. Their derivations are presented by using three methods, which are along the lines of the three types of well-known proofs of the q-Kummer summation theorem with a key role of the q-binomial theorem. In addition to the q-binomial theorem, the first proof makes use of Thomae’s q-integral representation and the second proof needs Heine’s transformation. Whereas the third proof utilizes only the q-binomial theorem. Subsequently, the applications of these summation formulas in obtaining the general contiguous extensions of the second and the third q-Kummer summation theorems are also presented. Furthermore, the investigated results are specialized to give many of the known as well as presumably new q-summation theorems, which are contiguous to the three q-Kummer summation theorems. This work is motivated by the observation that the basic (or q-) series and basic (or q-) polynomials, especially the basic (or q-) gamma and q-hypergeometric functions and basic (or q-) hypergeometric polynomials, are applicable particularly in several diverse areas including Number Theory, Theory of Partitions and Combinatorial Analysis as well as in the study of Combinatorial Generating Functions. Just as it is known in the theory of the Gauss, Kummer (or confluent), Clausen and the generalized hypergeometric functions, the parameters in the corresponding basic or quantum (or q-) hypergeometric functions are symmetric in the sense that they remain invariant when the order of the p numerator parameters or when the order of the q denominator parameters is arbitrarily changed. A case has therefore been made for the symmetry possessed not only by hypergeometric functions and basic or quantum (or q-) hypergeometric functions, which are studied in this paper, but also by the symmetric quantum calculus itself.

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Section: Introduction Motivation and Preliminariesmentioning

confidence: 99%

General Summation Formulas Contiguous to the q-Kummer Summation Theorems and Their Applications

et al. 2021

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A note on P-Q type modular equations and their evaluations

Shruthi

Kumar

2022

São Paulo J. Math. Sci.

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New proofs of some Dedekind η-function identities of level 6

Raksha,

Srivastava,

Sayinath

et al. 2023

Filomat

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Recently, Shaun Cooper proved several interesting ?-function identities of level 6 while finding series and iterations for 1/?. In this sequel, we present some new proofs of the ?-function identities of level 6 discovered by Cooper. Here, in this article, we make use of the modular equation of degree 3 in two methods. We further give some interesting combinatorial interpretations of colored partitions. We also briefly describe a potential direction for further researches based upon some related recent developments involving the Jacobi?s triple-product identity and the theta-function identities as well as on several other q-functions which emerged from the Rogers-Ramanujan continued fraction R(q) and its such associates as G(q) and H(q). We point out the importance of the usage of the classical q-analysis and we also expose the current trend of falsely-claimed ?generalization? by means of its trivial and inconsequential (p, q)-variation by inserting a forced-in redundant (or superfluous) parameter p.

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Some modular equations analogous to Ramanujan’s identities

Cited by 3 publications

References 10 publications

General Summation Formulas Contiguous to the q-Kummer Summation Theorems and Their Applications

General Summation Formulas Contiguous to the q-Kummer Summation Theorems and Their Applications

A note on P-Q type modular equations and their evaluations

New proofs of some Dedekind η-function identities of level 6

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