On Summation of p-Adic Series

Original · EN

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 ≤ 1 for 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 Aₖε (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.