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On certain extensions of Ramanujan's Master Theorem and their applications

Omprakash Atale, Mahendra Shirude

Abstract

S. Ramanujan introduced a technique in 1913 for providing analytic expressions for certain Mellin-type integrals which is now known as Ramanujan's Master Theorem. This technique was communicated through his "Quarterly Reports" and has a wide range of applicability in calculating the values of certain definite integrals. In this paper, we have presented some extensions of Ramanujan's Master Theorem that arise from the k-gamma function and the p-k gamma function. Furthermore, we have established a Mellin type double integral along with its corollaries. Some applications are established through the evaluation of a variety of definite integrals.

On certain extensions of Ramanujan's Master Theorem and their applications

Abstract

S. Ramanujan introduced a technique in 1913 for providing analytic expressions for certain Mellin-type integrals which is now known as Ramanujan's Master Theorem. This technique was communicated through his "Quarterly Reports" and has a wide range of applicability in calculating the values of certain definite integrals. In this paper, we have presented some extensions of Ramanujan's Master Theorem that arise from the k-gamma function and the p-k gamma function. Furthermore, we have established a Mellin type double integral along with its corollaries. Some applications are established through the evaluation of a variety of definite integrals.

Paper Structure

This paper contains 7 sections, 15 theorems, 111 equations.

Table of Contents

  1. Introduction
  2. Ramanujan's Master Theorem for k -gamma function
  3. Ramanujan’s Master Theorem for p-k gamma function
  4. Extensions of Ramanujan’s Master Theorem and their application to zeta function
  5. A double integral generalization
  6. Conclusion
  7. Acknowledgement

Key Result

Theorem 1.1

[Hardy-Ramanujan Master Theorem] Let $\varphi(z)$ be an analytic (single-valued) function, defined on a half-plane $H(\delta)=\{z \in {C}: \Re (z) \geq-\delta\}$ for some $0<\delta<1 .$ Suppose that, for some $A<\pi, \phi$ satisfies the growth condition $|\phi(v+i w)|<C e^{P v+A|w|}$ for all $z=v+i

Theorems & Definitions (32)

  • Theorem 1.1
  • proof
  • Theorem 2.1
  • proof
  • Corollary 2.1.1
  • proof
  • Corollary 2.1.2
  • proof
  • Corollary 2.1.3
  • proof
  • ...and 22 more