On Summation of p-Adic Series

arXiv: Number Theory, 2019

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 invesigated certain condition so that the p-adic series converges and gives rational sum for rational variable.

2019

The reason behind the essentiality of p-adic numbers is that it has has been very useful as a tool to solve many hard problems of pure mathematics specially on number theory. Over the last 3 decades, p-adic numbers has been successfuly applied into physical phenomenon as well, as formulated in the literature review paper [4]. Further, series plays a much crucial part in mathematics and physics. So it is important to investigate convergence and find out sum of series. However, we prefer to work with finite or rational values in physical measurement. Another important thing is that infinite series with rational numbers are rational numbers can be treated in any p-adic as well as in real number field, because rational numbers are endowed by real and p-adic norms. Specially, a real series which diverges in real case, needs p-adic investigation when its p-adic sum is a rational for a rational argument. We are mainly inspired by the works of the author of ([1], [2], [3]). In these works (...

Journal of Number Theory, 1992

P-adic Numbers, Ultrametric Analysis, and Applications, 2014

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 summable functional series with integer coefficients which contain factorials.

arXiv: Number Theory, 2018

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.

arXiv (Cornell University), 2004

Some p-adic series with factorials are considered.

2021

The aim of this paper is to construct generating functions for some families of special finite sums with the aid of the Newton-Mercator series, hypergeometric series, and p-adic integral (the Volkenborn integral). By using these generating functions, their functional equations, and their partial derivative equations, many novel computational formulas involving the special finite sums of (inverse) binomial coefficients, the Bernoulli type polynomials and numbers, Euler polynomials and numbers, the Stirling numbers, the (alternating) harmonic numbers, the Leibnitz polynomials and others. Among these formulas, by considering a computational formula which computes the aforementioned certain class of finite sums with the aid of the Bernoulli numbers and the Stirling numbers of the first kind, we present a computation algorithm and we provide some of their special values. Morover, using the aforementioned special finite sums and combinatorial numbers, we give relations among multiple alte...

2018

In this paper we have discussed about the region of convergence of a power series in p-adic field. We have investigated some sufficient conditions for which a power series has same radius of convergence with respect to both p-adic absolute value and usual absolute value on the field of rational numbers Q i.e., both in p-adic field Qp and real field R. Finally we have characterised the rationals for which the power series is convergent with respect to usual absolute value and p-adic absolute value within the same radius of convergence.

Journal of Number Theory, 1989

An algorithm is introduced and shown to lead to various unique series expansions of p-adic numbers, as sums of rational numbers. The degrees of approximation by the partial sums of these series are investigated. '(" 1989 Academic Press, Inc.

Symmetry, 2021

The aim of this paper is to study and investigate generating-type functions, which have been recently constructed by the author, with the aid of the Euler’s identity, combinatorial sums, and p-adic integrals. Using these generating functions with their functional equation, we derive various interesting combinatorial sums and identities including new families of numbers and polynomials, the Bernoulli numbers, the Euler numbers, the Stirling numbers, the Daehee numbers, the Changhee numbers, and other numbers and polynomials. Moreover, we present some revealing remarks and comments on the results of this paper.