On p-Adic Approximation of Sums of Binomial Coefficients

Abstract

We propose higher-order generalizations of Jacobsthal's p-adic approximation for binomial coefficients. Our results imply explicit formulae for linear combinations of binomial coefficients ip p (i = 1, 2, . . . ) that are divisible by arbitrarily large powers of prime p.

A Generalization of $p$-Adic Factorial

“…Both of Robert and Menken have used the p-adic factorial only to define the p-adic gamma function, without giving its properties. Furthermore, Aidagulov and Alekseyev in [6] have also used the so-called modified (p-adic) factorial, with the notation n! p , to study the modified (p-adic) binomial coefficients.…”