On Rational Invariant Summation of p-Adic Power Series with Binomial Coefficient

TL;DR

This paper investigates p-adic power series with binomial coefficients, deriving an invariant summation formula for rational arguments, and applies it to establish new relations involving Bernoulli numbers and polynomials.

Contribution

It introduces a new invariant summation formula for p-adic series with binomial coefficients and explores conditions for rational invariant sums.

Findings

Derived a recurrence relation for p-adic series

Established conditions for convergence to rational sums

Applied the formula to Bernoulli numbers and polynomials

Abstract

In the work we have considered p-adic functional series with binomial coefficients and discussed its p-adic convergence. Then we have derived a recurrence relation following with a summation formula which is invariant for rational argument. More precisely, we have investigated a sufficient condition under which the p-adic power series converges to a rational invariant sum 0. Finally we have shown application of the invariant summation formula to get some intersting relations involving Bernoulli numbers and Bernoulli polynomials.