A series representation of Euler–Stieltjes constants and an identity of Ramanujan

Abstract

We derive a series representation of the generalized Stieltjes constants which arise in the Laurent series expansion of partial zeta function at the point $s=1$. In the process, we introduce a generalized gamma function and deduce its properties such as functional equation, Weierstrass product and reflection formulas along the lines of the study of a generalized gamma function introduced by Dilcher in 1994. These properties are used to obtain a series representation for the $k$-th derivative of Dirichlet series with periodic coefficients at the point $s=1$. Another application involves evaluation of a class of infinite products of which a special case is an identity of Ramanujan.