Extended higher Herglotz functions I. Functional equations

In 1975, Don Zagier obtained a new version of the Kronecker limit formula for a real quadratic field which involved an interesting function $F\left(x\right)$ which is now known as the Herglotz function. As demonstrated by Zagier, and very recently by Radchenko and Zagier, $F\left(x\right)$ satisfies beautiful properties which are of interest in both algebraic number theory as well as in analytic number theory. In this paper, we study $F_{k,N}\left(x\right)$, an extension of the Herglotz function which also subsumes higher Herglotz function of Vlasenko and Zagier. We call it the extended higher Herglotz function. It is intimately connected with a certain generalized Lambert series as well as with a generalized cotangent Dirichlet series inspired from Krätzel's work. We derive two different kinds of functional equations satisfied by $F_{k,N}\left(x\right)$. Radchenko and Zagier gave a beautiful relation between the integral $\underset{0}{\overset{1}{\int }}\frac{\log (1+t^x)}{1+t}dt$ and $F\left(x\right)$ and used it to evaluate this integral at various rational as well as irrational arguments. We obtain a relation between $F_{k,N}\left(x\right)$ and a generalization of the above integral involving polylogarithm. The asymptotic expansions of $F_{k,N}\left(x\right)$ and some generalized Lambert series are also obtained along with other supplementary results.