On a function of Ramanujan twisted by a logarithm

TL;DR

This paper derives a new two-term functional equation involving the digamma function and logarithms, utilizing Ramanujan's modular relations and Deninger's function derivatives, with a focus on an integral of independent interest.

Contribution

It introduces a novel functional equation for an infinite series with digamma and logarithmic factors, connecting Ramanujan's modular relations and Deninger's functions.

Findings

Derived a two-term functional equation involving digamma and logarithmic functions.

Connected Ramanujan's modular relation with recent results on Deninger's function.

Provided multiple representations of an integral function of independent interest.

Abstract

A two-term functional equation for an infinite series involving the digamma function and a logarithmic factor is derived. A modular relation on page 220 of Ramanujan's Lost Notebook as well as a corresponding recent result for the derivative of Deninger's function are two main ingredients in its derivation. An interesting integral $\mathscr{H}(x)$, which is of independent interest, plays a prominent role in our functional equation. Several alternative representations for $\mathscr{H}(x)$ are obtained.