On summation of p-adic series

Abstract

© 2018 American Mathematical Society. Summation of the p-adic functional series∑ εn n! Pkε(n;x)xn, where Pkε(n; x) is a polynomial in x and n with rational coefficients, and ε = ±1, is considered. The series is convergent in the domain |x|p ≤ 1for all primes p. It is found the general form of polynomials Pkε (n; x) which provide rational sums when x ∈ Z. A class of generating polynomials Aεk(n; x) plays a central role in the summation procedure. These generating polynomials are related to many sequences of integers. This is a brief review with some new results.