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In this paper we have discussed convergence of power series both in p-adic norm as well as real norm. We have investigated rational summability of power series with respect to both p-adic norm and real norm under certain conditions. Then we have studied convergence of specially constructed power series and derived summation formula. Finally, we have studied the adele, idele and some results regarding it with the help of convergent power series.

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Power series are introduced that are simultaneously convergent for all real and all 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∈Q adeles of series are constructed. Possible applications at the Planck scale are also considered.

A brief review of some selected topics in p-adic mathematical physics is presented. Comment: 36 pages

An introduction to p-adic numbers and p-adic analysis

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URL:https://www.math.uconn.edu/∼kconrad/blurbs/gradnumthy/infseriespadic.pdf

On some p-adic series with factorials, arXiv preprint math-ph/0402050

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Dragovich, B., On some p-adic series with factorials, arXiv preprint math-ph/0402050, 2004.

E-mail address: 1 aask2003@yahoo.co.in, aashaikh@math.buruniv.ac

- A M Robert

Robert, A. M., A course in p-adic analysis, Springer-Verlag, 2000.
Department of Mathematics,, The University of Burdwan,,
Burdwan-713101, West Bengal, India.
E-mail address: 1 aask2003@yahoo.co.in, aashaikh@math.buruniv.ac.in
Department of Mathematics, The University of Burdwan, Burdwan-713101, India.
E-mail address: 2 mabudji@gmail.com