Novel integral representations of the Riemann zeta-function and Dirichlet eta-function, close expressions for Laurent series expansions of powers of trigonometric functions and digamma function, and summation rules

TL;DR

This paper introduces new integral representations for the Riemann zeta and Dirichlet eta functions, enabling explicit Laurent series expansions of related functions and deriving new summation rules for special values and constants.

Contribution

It provides novel integral formulas linking zeta and eta functions with trigonometric and digamma functions, leading to explicit series expansions and summation rules.

Findings

New integral representations for zeta and eta functions.

Explicit Laurent series expansions for powers of trigonometric and digamma functions.

Derivation of new summation rules for special constants and values.

Abstract

We have established novel integral representations of the Riemann zeta-function and Dirichlet eta-function based on powers of trigonometric functions and digamma function, and then use these representations to find close forms of Laurent series expansions of these same powers of trigonometric functions and digamma function. The so obtained series can be used to find numerous summation rules for certain values of the Riemann zeta and related functions and numbers, such as e.g. Bernoulli and Euler numbers.