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) = ₀^1 (1-utᶻ) v^{-1-t} dt, for \, \, \, \, Re (z) gt;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) amp;=₀¹ (1-utᶻ) (1-wtᶻ) v^{-1-t}dt, \ₖ (z;u, v) amp;= ₀^1 ᵏ (1-utᶻ) v^{-1-t} \, dt, align*for Re (z) gt;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.