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


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 D n (x) by means of the following exponential generating function: In the case x = 0 in the Eq.(1.1), one can get D n (0) := D n 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 ord p a be the highest power of p that divides a, i.e., the greatest m such that a 0 (mod p m ) where we used the notation a b (mod c) meant c divides a b.Note that ord p 0 = 1.The p-adic absolute value (norm) of x is given by jxj p = p ordpx for x 6 = 0 and j0j p = 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, Q p = 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 n 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 C p with jq 1j p < p 1 p 1 so that q x = exp (x log q) for jxj p 5 1.
For f 2 U D (Z p ) = ff jf is uniformly di¤erentiable function at a point a 2 Z p g, Kim de…ned the q-Volkenborn integral or p-adic q-integral on Z p of a function f 2 U D (Z p ) in [10] as follows: . Then, we see that For these related issues, see [1; 3; 9-11; 17] and related references cited therein.The q-Daehee numbers D n;q and q-Daehee polynomials D n;q (x) are de…ned by means of q-Volkenborn integrals: where the symbol (x) n denotes the falling factorial given by The falling factorial (x) n satis…es the following identity: where S 1 (n; k) denotes the Stirling number of the …rst kind (see [9; 11]).
It is obvious that lim q!1 D n;q := D n and lim q!1 D n;q (x) := D n (x).
The q-Daehee numbers and polynomials of the second kind are introduced by the following q-Volkenborn integrals: The q-Daehee polynomials and numbers and their various generalizations have been studied by many mathematicians, cf.[1; 9; 11]; see also the related references cited therein.The p-adic gamma function is de…ned as follows where n approaches x through positive integers.
The p-adic Euler constant p is de…ned by the following formula The p-adic gamma function in conjunction with its several extensions and p-adic Euler constant have been developed by many physicists and mathematicians, cf.[2; 4-8; 12; 14-16 ; 18]; see also the references cited in each of these earlier works.For x 2 Z p , the symbol x n is given by x 0 = 1 and x n = x(x 1) (x n+1) n!
Proposition 1. (Kim et al. [9]) The following relation holds true for n 0: ( 1) j q n j j (1 q) n j . (1.12) Let x 2 Z p and n 2 N. The functions x !x n form an orthonormal base of the space C (Z p ! C p ) with respect to the euclidean norm k k 1 .The mentioned orthonormal base satisfy the following equality: (see [13; 16; 18]) . (1.13) Mahler investigated a generalization for continuous maps of a p-adic variable utilizing the special polynomials as binomial coe¢ cient polynomial [13] in 1958.It implies that for any f 2 C (Z p ! C p ), there exist unique elements a o ; a 1 ; a 2 ; : : : of C p such that The base n : n 2 N is named as Mahler base of the space C (Z p ! C p ) ; and the components The Mahler expansion of the p-adic gamma function p and its Mahler coe¢ cients are discovered in [16] as follows.
Proposition 2. For x 2 Z p , let p (x + 1) = P 1 n=0 a n x n be Mahler series of p .Then its coe¢ cients satisfy the following identity: The outlines of this paper are as follows: the …rst part is introduction which provides the required information, notations, de…nitions and motivation; in part 2, we are interested in constructing the correlations between the p-adic gamma function and the q-Daehee polynomials and numbers by using the methods of the q-Volkenborn integral and Mahler series expansion; in the last part, we examine the results derived in this paper.

Main Results
This section provides some properties, identities and correlations for the mentioned gamma function, q-Daehee polynomials and numbers, Stirling numbers of the …rst kind and p-adic Euler constant.
The q-Volkenborn integral on Z p of the p-adic gamma function via Proposition 1 and Proposition 2 is as follows.
Theorem 1.The following identity holds true for n 2 N: where a n is given by Proposition 2.
Proof.For x; y 2 Z p , by Proposition 2, we get and using the formula (1.12), we acquire which implies the desired result.
We present one other q-Volkenborn integral of the p-adic gamma function via q-Daehee polynomials by Theorem 2.
Theorem 2. Let x; y 2 Z p .We have where a n is given by Proposition 2.
Proof.For x; y 2 Z p , by the relation and Proposition 2, we get which is the desired result (2.1) via Eq.( We now examine a consequence of the Theorem 2 as follows.
Corollary 1. Choosing y = 0 in Theorem 2 gives the fo llowing relation including p and D n;q : where a n is given by Proposition 2.
Here is the p-adic q-integral of the derivative of the p-adic gamma function.
Theorem 3.For x; y 2 Z p ; we have n j 1 D j;q (y) (n j) j! .
Proof.In view of (1.3), we obtain and using (1.13), we derive n j 1 n j D n;q (y) j! : The immediate result of Theorem 3 is given as follows.
n j 1 D j;q (n j) j! .
The p-adic gamma function can be determined by means of the Stirling numbers of the …rst kind as follows.
Theorem 4. For x; y 2 Z p ; we obtain Proof.From (1.7) and Proposition 2, we have which gives the desired result.
As a result of Theorem 4, one other q-Volkenborn integral of the p-adic gamma function via the q-Bernoulli polynomials is stated below.Corollary 3.For x; y 2 Z p ; we acquire where B n (x) denotes n-th q-Bernoulli polynomial de…ned in [9] by the following p-adic q-integral on Z p : We now provide a new and interesting representation of the p-adic Euler constant by means of q-Daehee polynomials and numbers.
Proof.Taking f (x) = 0 p (x) in Eq. (1.3) yields the following result q Z Zp 0 p (x + 1) d q (x) Z Zp 0 p (x) d q (x) = (q 1) 0 p (0) + q 1 log q (2) where p (0) is the second derivative of the p-adic gamma function at x = 0, and with some basic calculations and using Theorem 3, we have

x = P 1
n= k a n p n : 0 5 a i 5 p 1 denotes the …eld of all p-adic numbers, Z p = n x 2 Q p : jxj p 5 1 o denotes the ring of all p-adic integers and C p denotes the completion of the algebraic closure of Q p .