Summation of certain trigonometric series with logarithmic coefficients

TL;DR

This paper derives sums of specific trigonometric series with logarithmic coefficients by extending Lerch's approach, evaluating related integrals with hyperbolic functions using Frullani's theorem, and expressing results via the digamma function.

Contribution

It introduces a novel extension of Lerch's method to evaluate certain trigonometric series with logarithmic coefficients and connects these sums to integrals involving hyperbolic functions and the digamma function.

Findings

Explicit formulas for the sums of the series are obtained.

Two non-trivial integrals involving hyperbolic functions are evaluated.

Results are expressed in terms of the digamma function.

Abstract

The sums of three trigonometric series with logarithmic coefficients are derived by extending an approach first utilized by Lerch. By applying Frullani's theorem to two of these series, two non-trivial integrals involving hyperbolic functions are evaluated in terms of the digamma function.