On Summation of $p$-Adic Series

Abstract

Summation of the $p$-adic functional series $\sum {\epsilon }^{n}n!P_k^{\epsilon }(n;x)x^n,$ where $P_k^{\epsilon }(n;x)$ is a polynomial in $x$ and $n$ with rational coefficients, and $\epsilon =\pm 1$, is considered. The series is convergent in the domain $|x{|}_p\le 1$ for all primes $p$. It is found the general form of polynomials $P_k^{\epsilon }(n;x)$ which provide rational sums when $x\in \mathbb{Z}$. A class of generating polynomials $A_k^{\epsilon }(n;x)$ 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.

14 pages, accepted for publication in proceedings of 14th International Conference on p-Adic Functional Analysis (Contemporary Mathematics, AMS)

References in corpus (2)

• On Some Finite Sums with Factorials
• On p-Adic Power Series

Cited by in corpus (1)

• Irrationality of the sum of a $p$-adic series