Analogues of Herglotz-Zagier-Novikov function

Abstract

Recently, Choie and Kumar extensively studied the Herglotz-Zagier-Novikov function F(z

u, v), defined as ?1 F(z

u, v) = 0 log(1 ? utz ) v?1 dt, for Re(z) > 0. ? t They obtained two-term, three-term and six-term functional equations for F(z

u, v) and also evaluated special values in terms of di-logarithmic functions. Motivated from their work, we study the following two integrals, ?1 log(1 ? utz ) log(1 ? wtz ) F(z

u, v, w) = 0 v?1 dt, ? t ?1 logk (1 ? utz ) Fk (z

u, v) = 0 v?1 dt, ? t for Re(z) > 0 and k ? N. For k = 1, the integral Fk (z

u, v) reduces to F(z

u, v). This allows us to recover the properties of F(z

u, v) by studying the properties of Fk (z

u, v). We evaluate special values of these two functions in terms of poly-logarithmic functions. � 2025, Hardy-Ramanujan Society. All rights reserved.