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Generalization of some of Ramanujan's formulae

Aung Phone Maw

TL;DR

The paper develops a systematic partial-fraction method to generalize Ramanujan's identities for odd zeta values and related modular objects, notably yielding a general transformation formula that covers the eta-function transformation as a special case. It proves Propositions 1–4 by manipulating elementary partial-fraction decompositions and Mittag-Leffler expansions, then demonstrates how these yield Ramanujan's $\zeta(2n+1)$ identities and a broad, transferable framework for generating new identities. A central result is a transformation formula that subsumes Ramanujan's log-eta relations and yields numerous corollaries under various parameter choices. Extending the method, the authors develop a general approach to generalized Lambert-series, connecting zeta values, Bernoulli numbers, and theta/eta-type transformations within a single, versatile framework. The work provides a comprehensive toolkit for constructing rapidly convergent series identities in analytic number theory and modular forms.

Abstract

We shall make use of the method of partial fractions to generalize some of Ramanujan's infinite series identities, including Ramanujan's famous formula for $ζ(2n+1)$, and we shall also give a generalization of the transformation formula for the Dedekind eta function. It is shown here that the method of partial fractions can be used to obtain many similar identities of this kind.

Generalization of some of Ramanujan's formulae

TL;DR

The paper develops a systematic partial-fraction method to generalize Ramanujan's identities for odd zeta values and related modular objects, notably yielding a general transformation formula that covers the eta-function transformation as a special case. It proves Propositions 1–4 by manipulating elementary partial-fraction decompositions and Mittag-Leffler expansions, then demonstrates how these yield Ramanujan's identities and a broad, transferable framework for generating new identities. A central result is a transformation formula that subsumes Ramanujan's log-eta relations and yields numerous corollaries under various parameter choices. Extending the method, the authors develop a general approach to generalized Lambert-series, connecting zeta values, Bernoulli numbers, and theta/eta-type transformations within a single, versatile framework. The work provides a comprehensive toolkit for constructing rapidly convergent series identities in analytic number theory and modular forms.

Abstract

We shall make use of the method of partial fractions to generalize some of Ramanujan's infinite series identities, including Ramanujan's famous formula for , and we shall also give a generalization of the transformation formula for the Dedekind eta function. It is shown here that the method of partial fractions can be used to obtain many similar identities of this kind.
Paper Structure (5 sections, 20 theorems, 91 equations)

This paper contains 5 sections, 20 theorems, 91 equations.

Table of Contents

  1. Introduction
  2. Derivations
  3. A certain transformation formula
  4. Some further consequences
  5. On a general series

Key Result

Proposition 1

For $|\theta_i| \leq \pi$, $1 \leq i \leq M$ and arbitrary $t$, and if for complex values $a_1,...,a_M$ and nonzero complex values $x_1,...,x_M$ such that for all $i \neq j$, and for all natural numbers $n$ and $k$, we have $x_i^2n^2-x_j^2k^2 \neq a_j^2-a_i^2$, and if for each $i \neq j$, $\textbf{I

Theorems & Definitions (20)

  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Proposition 6
  • Proposition 7
  • Corollary 8
  • Corollary 9
  • Corollary 10
  • ...and 10 more