Analogues of Harglotz-Zagier-Novikov function

TL;DR

This paper extends the study of the Herglotz-Zagier-Novikov function by introducing and analyzing new related integrals involving powers and products of logarithms, deriving their properties and special values in terms of polylogarithms.

Contribution

The paper introduces generalized integrals related to the Herglotz-Zagier-Novikov function and derives their properties and special values, expanding the understanding of these functions.

Findings

Derived functional equations for the new integrals.

Evaluated special values in terms of polylogarithmic functions.

Connected properties of generalized integrals to the original function.

Abstract

Recently, Choie and Kumar extensively studied the Herglotz-Zagier-Novikov function $\mathfrak{F}(z;u,v)$, defined as \begin{align*} \mathfrak{F}(z;u,v) = \int_{0}^{1} \frac{\log(1-ut^z)}{v^{-1}-t} dt, \quad \textrm{for} \quad \mathfrak{Re}(z)>0. \end{align*} They obtained two-term, three-term and six-term functional equations for $\mathfrak{F}(z;u,v)$ and also evaluated special values in terms of di-logarithmic functions. Motivated from their work, we study the following two integrals, \begin{align*} \mathfrak{F}(z;u,v,w) &=\int_{0}^1 \frac{\log(1-ut^z)\log(1-wt^z)}{v^{-1}-t}\text{d}t, \\ \mathfrak{F}_k(z;u,v) &= \int_{0}^{1} \frac{\log^k(1-ut^z)}{v^{-1}-t} \, \text{d}t, \end{align*} for $\mathfrak{Re}(z)>0$ and $k \in \mathbb{N}$. For $k=1$, the integral $\mathfrak{F}_k(z;u,v)$ reduces to $\mathfrak{F}(z;u,v)$. This allows us to recover the properties of $\mathfrak{F}(z;u,v)$ by studying the properties of $\mathfrak{F}_k(z;u,v)$. We evaluate special values of these two functions in terms of poly-logarithmic functions.