Finite Lerch Functions

Abstract

We define the finite Lerch function

$H_n ^{(s)}(x,y)=\sum^{n-1}_{k=0}\frac{x^k}{(k+y)^s}$.

Then we explore some properties of them and give some applications, including

Coppo's formula, the sum of powers of consecutive integers, Mellin

differential operator, and some others.