SOME RELATIONSHIPS INCLUDING p-ADIC GAMMA FUNCTION AND q-DAEHEE POLYNOMIALS AND NUMBERS

In this paper, we investigate p-adic q-integral (q-Volkenborn integral) on Zp of p-adic gamma function via their Mahler expansions. We also derived two q-Volkenborn integrals of p-adic gamma function in terms of q-Daehee polynomials and numbers and q-Daehee polynomials and numbers of the second kind. Moreover, we discover q-Volkenborn integral of the derivative of p-adic gamma function. We acquire the relationship between the p-adic gamma function and Stirling numbers of the …rst kind. We …nally develop a novel and interesting representation for the p-adic Euler constant by means of the q-Daehee polynomials and numbers. 1. Introduction The p-adic numbers are a counterintuitive arithmetic system, which were …rstly introduced by the Kummer in 1850. In conjunction with the introduction of these numbers, some mathematicians and physicists started to investigate new scienti…c tools utilizing their useful and positive properties. Firstly Kurt Hensel, the German mathematician, (1861-1941) improved the p-adic numbers in a study concerned with the development of algebraic numbers in power series in circa 1897. Some e¤ects of these researches have emerged in mathematics and physics such as p-adic analysis, string theory, p-adic quantum mechanics, QFT, representation theory, algebraic geometry, complex systems, dynamical systems, genetic codes and so on (cf. [1-10; 12-18]; also see the references cited in each of these earlier studies). The one important tool of these investigations is p-adic gamma function which is …rstly described by Yasou Morita [15] in about 1975. Intense research activities in such an area as p-adic gamma function is principally motivated by their importance in p-adic analysis. Therefore, in recent fourty years, p-adic gamma function and its generalizations have been investigated and studied extensively by many mathematicians, cf. [2; 4-8; 12; 14-16; 18]; see also the related references cited therein. Kim et al. [11] de…ned Daehee polynomials Dn(x) by means of the following exponential generating function: 1 X n=0 Dn(x) t n! = log (1 + t) t (1 + t) x . (1.1) In the case x = 0 in the Eq. (1.1), one can get Dn(0) := Dn standing for n-th Daehee number, see [1; 9; 11] for more detailed information about these related issues. Let p 2 f2; 3; 5; 7; 11; 13; 17; g be a prime number. For any nonzero integer a, let ordpa be the highest power of p that divides a, i.e., the greatest m such that a 0 (mod p) where we used the notation a b (mod c) meant c divides a b. Note that ordp0 = 1. The p-adic absolute value (norm) of x is given by jxjp = p ordpx for x 6= 0 and j0jp = 0: Now we provide some basic notations: N = f1; 2; 3; g denotes the set of all natural numbers, Z = f ; ; 1; 0; 1; g denotes the ring of all integers, C denotes the …eld of all complex numbers, Qp = x = P1 n= k anp n : 0 5 ai 5 p 1 denotes the …eld of all p-adic numbers, Zp = n x 2 Qp : jxjp 5 1 o denotes the ring of all p-adic integers and Cp denotes the completion of the algebraic closure of Qp. 1991 Mathematics Subject Classi…cation. Primary 05A10, 05A30; Secondary 11B65, 11S80, 33B15. Key words and phrases. p-adic numbers, p-adic gamma function, p-adic Euler constant, q-Daehee polynomials, Stirling numbers of the …rst kind. Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 19 February 2018 doi:10.20944/preprints201802.0118.v1 © 2018 by the author(s). Distributed under a Creative Commons CC BY license. 2 U. Duran and M. Acikgoz For more information about p-adic analysis, see [1-10; 12-18] and related references cited therein. The q-number is de…ned by [n]q = q 1 q 1 . The symbol q can be variously considered as indeterminates, complex number q 2 C with 0 < jqj < 1, or p-adic number q 2 Cp with jq 1jp < p 1 p 1 so that q = exp (x log q) for jxjp 5 1. For f 2 UD (Zp) = ff jf is uniformly di¤erentiable function at a point a 2 Zpg, Kim de…ned the qVolkenborn integral or p-adic q-integral on Zp of a function f 2 UD (Zp) in [10] as follows: Iq(f) = Z Zp f (x) d q (x) = lim N!1 1 [pN ]q p 1 X