Power series everywhere convergent on R and all Q_p

TL;DR

This paper introduces power series that converge across all real and p-adic numbers, enabling new rational-sum series and adelic constructions with potential applications in physics at the Planck scale.

Contribution

It presents novel power series convergent everywhere on R and Q_p, and develops new summable series with rational sums and adelic series for rational arguments.

Findings

Power series convergent on all R and Q_p.

New series with rational sums derived from factorial structures.

Construction of adelic series for rational arguments.

Abstract

Power series are introduced that are simultaneously convergent for all real and p-adic numbers. Our expansions are in some aspects similar to those of exponential, trigonometric, and hyperbolic functions. Starting from these series and using their factorial structure new and summable series with rational sums are obtained. For arguments $x\in {\mathbb Q}$ adeles of series are constructed. Possible applications at the Planck scale are also considered.