Power series everywhere convergent on R and all Q_p

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.