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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 particularly, we have investigated certain condition so that the p-adic series converges and gives rational sum for rational variable.

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

Summation of a large class of the functional series, which terms contain
factorials, is considered. We first investigated finite partial sums for
integer arguments. These sums have the same values in real and all p-adic
cases. The corresponding infinite functional series are divergent in the real
case, but they are convergent and have p-adic invariant sums in p-adic cases.
We found polynomials which generate all significant ingredients of these series
and make connection between their real and p-adic properties. In particular, we
found connection of one of our integer sequences with the Bell numbers.

We consider summation of some finite and infinite functional p-adic series
with factorials. In particular, we are interested in the infinite series which
are convergent for all primes p, and have the same integer value for an integer
argument. In this paper, we present rather large class of such p-adic
functional series with integer coefficients which contain factorials. By
recurrence relations, we constructed sequence of polynomials A_k(n;x) which are
a generator for a few other sequences also relevant to some problems in number
theory and combinatorics.

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.

p-adic invariant summation of some p-adic Functional series, p-adic numbers, ultrametric analysis and applications

- B Dragovich

B. Dragovich, p-adic invariant summation of some p-adic Functional series, p-adic numbers, ultrametric
analysis and applications, 2014.

On summation of p-adic series, communications in contemporary mathematics

- B Dragovich

B. Dragovich, On summation of p-adic series, communications in contemporary mathematics, 2017.

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

- B Andrew

Andrew, B., An introduction to p-adic numbers and p-adic analysis, URL:https://www.maths.gla.ac.uk/
∼ajb.

Infinite series in p-adic fields

- K Conrad

Conrad, K., Infinite series in p-adic fields,
URL:https://www.math.uconn.edu/∼kconrad/blurbs/gradnumthy/infseriespadic.pdf

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

- B Dragovich

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

On p-adic mathematical physics

- B Dragovich
- A Y Khrennikov
- S V Kozyrev
- I V Volvich

B. Dragovich, AY. Khrennikov, SV. Kozyrev, IV. Volvich, "On p-adic mathematical physics", p-adic numbers, p-adic analysis and applications, 2009.

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

- E Kummer

E. Kummer, Uber die Ergnzungsstze zu den allgemeinen Reciprocittsgesetzen, Journal fr die reine und angewandte Mathematik, 1852.
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