Analogues of Herglotz-Zagier-Novikov function

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