An analogue of Herglotz-Zagier-Novikov function

Abstract

In mathematics, evaluating an integral in terms of well-known constants is always a fascinating and challenging task. Recently, Choie and Kumar [1] extensively studied the Herglotz-Zagier-Novikov function F(z; u, v). It is defined as the following integral: F(z; u, v) := Z 1 0 log(1 − utz) v−1 − t dt, for Re(z) > 0, (0.1) where u ∈ L and v ∈ L′. 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, in this thesis, we study the following two integrals, for Re(z) > 0, and any natural number k, F(z; u, v,w) := Z 1 0 log(1 − utz) log(1 − wtz) v−1 − t dt, (0.2) Fk(z; u, v) := Z 1 0 logk(1 − utz) v−1 − t dt, (0.3) where u ∈ L and v ∈ L′. For k = 1, the above integral (0.3) reduces to (0.1). This allows to recover the properties of F(z; u, v) by studying the properties of Fk(z; u, v). One of the main aims of this thesis is to evaluate special values of these two integrals in terms of poly-logarithmic functions.