On p-adic Diamond–Euler Log Gamma functions

Abstract

Abstract Text In this paper, using the fermionic p-adic integral on Z p , we define the corresponding p-adic Log Gamma functions, so-called p-adic Diamond–Euler Log Gamma functions. We then prove several fundamental results for these p-adic Log Gamma functions, including the Laurent series expansion, the distribution formula, the functional equation and the reflection formula. We express the derivative of p-adic Euler L-functions at s = 0 and the special values of p-adic Euler L-functions at positive integers as linear combinations of p-adic Diamond–Euler Log Gamma functions. Finally, using the p-adic Diamond–Euler Log Gamma functions, we obtain the formula for the derivative of the p-adic Hurwitz-type Euler zeta function at s = 0 , then we show that the p-adic Hurwitz-type Euler zeta functions will appear in the studying for a special case of p-adic analogue of the ( S , T ) -version of the abelian rank one Stark conjecture. Video For a video summary of this paper, please click here or visit http://youtu.be/DW77g3aPcFU .