The Boole polynomials associated with the $p$-adic gamma function

Ugur Duran, Mehmet Acikgoz

We set some correlations between Boole polynomials and $p$-adic gamma function in conjunction with $p$-adic Euler contant. We develop diverse formulas for $p$-adic gamma function by means of their Mahler expansion and fermionic $p$-adic integral on $\mathbb Z_p$. Also, we acquire two fermionic $p$-adic integrals of $p$-adic gamma function in terms of Boole numbers and polynomials. We then provide fermionic $p$-adic integral of the derivative of $p$-adic gamma function and a representation for the $p$-adic Euler constant by means of the Boole polynomials. Furthermore, we investigate an explicit representation for the aforesaid constant covering Stirling numbers of the first kind.