On a function of Ramanujan twisted by a logarithm

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.