Analogues of Herglotz-Zagier-Novikov function

Recently, Choie and Kumar extensively studied the Herglotz-Zagier-Novikov function F(z;u,v), defined as {align*}F(z;u,v) = ∫₀¹ {log(1-ut^z)}{v⁻¹-t} dt, for \,\,\,\, Re(z)>0.{align*}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, {align*}F(z;u,v,w) &=∫₀^1 {log(1-ut^z)log(1-wt^z)}{v⁻¹-t}dt, _k(z;u,v) &= ∫₀¹ {log^k(1-ut^z)}{v⁻¹-t} \, dt,{align*}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.