A note on <i>p</i>-adic Raabe formulas

Abstract

The Hurwitz-type Euler (or the alternating Hurwitz) zeta function is defined by the following series ς_E(s,?) = ∑ for Re(s) > 0 and ? ∈ ℂ \ (− ∞, 0]: It can be analytically continued to the entire s-plane without any pole. In this note, we give a new proof of an analogue of Raabe's 1843 formula for p-adic Hurwitz-type Euler zeta functions by modifying Cohen and Friedman's method. We also use this method to construct the Raabe formulas for the nth derivative of p-adic Diamond-Euler Log Gamma functions. In addition, we obtain some identities for the fermionic p-adic integration, from which, some classical formulas for Euler polynomials have been deduced.