On a $p$-Adic Interpolating Function for the Generalized Genocchi Numbers and Its Behaviour at $s=0$

Abstract

In this work, we consider the generalized Genocchi numbers and polynomials. However, we introduce an analytic interpolating function for the generalized Genocchi numbers attached to $\chi$ at negative integers in the complex plane, and also we define the Genocchi $p$-adic $L$-function. As a result, we derive the value of the partial derivative of the Genocchi $p$-adic $l$-function at $s=0$.