Analogues of Harglotz-Zagier-Novikov function
Diksha Rani Bansal, Bibekananda Maji, Pragya Singh
TL;DR
The paper extends the Herglotz-Zagier-Novikov framework by introducing $\mathcal{F}(z;u,v,w)$ and $\mathcal{F}_k(z;u,v)$, establishing generalized duplication and multiplication formulas, analytic continuation, and explicit evaluations at rational args in terms of logs and polylogarithms. It builds on CK2023 results for $\mathcal{F}(z;u,v)$ and leverages polylog identities, including Rogers’ dilogarithm relation, to derive closed-form expressions such as $\mathcal{F}(1;u,u,u)=-\frac{1}{3}\log^3(1-u)$ and a comprehensive formula for $\mathcal{F}(1;u,v,w)$. The work provides practical schemes to compute $\mathcal{F}(p/q;u,v,w)$ via sums of $\mathcal{F}(1;\cdot)$ terms and delivers explicit $\mathcal{F}_k(\frac{1}{n};u,v)$ formulas, including the $u=v$ special case. These results deepen the understanding of generalized Kronecker-limit-type integrals and their connections to polylogarithmic constants with potential applications in Stark-type contexts.
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.