Analysis of the p-adic q-Volkenborn Integrals: an approach to Apostol-type special numbers and polynomials

By applying the p-adic q-Volkenborn Integrals including the bosonic and the fermionic p-adic integrals on p-adic integers, we define generating functions, attached to the Dirichlet character, for the generalized Apostol-Bernoulli numbers and polynomials, the generalized Apostol-Euler numbers and polynomials, generalized Apostol-Daehee numbers and polynomials, and also generalized Apostol-Changhee numbers and polynomials. We investigate some properties of these numbers and polynomials with their generating functions. By using these generating functions and their functional equation, we give some identities and relations including the generalized Apostol-Daehee and Apostol-Changhee numbers and polynomials, the Stirling numbers, the Bernoulli numbers of the second kind, Frobenious-Euler polynomials, the generalized Bernoulli numbers and the generalized Euler numbers and the Frobenious-Euler polynomials. By using the bosonic and the fermionic p-adic integrals, we derive integral represantations for the generalized Apostol-type Daehee numbers and the generalized Apostol-type Changhee numbers.


PUBLIC INTEREST STATEMENT
Recently, the special numbers and polynomials have been studied by many authors because these numbers and polynomials have been used in many branches of pure and applied mathematics, physics, engineering and computer science. Polynomials are used in elementary word problems to complicated problems in the sciences, approximate or curve fit experimental data, calculate beam deflection under loading, represent some properties of gases, and perform computer-aided geometric design in engineering; moreover, polynomials are used as solutions of differential equations. Using the p-adic q-Volkenborn integrals as a novel method, we construct new families of generalized Apostol-type special numbers and polynomials attached to the Dirichlet character. We give relations between these polynomials and some well-known special numbers and polynomials such as the Bernoulli and Euler polynomials, Bernstein polynomials, Peters polynomials, the Boole polynomials, the generalized Apostol-type numbers and polynomials, and the Stirling numbers.

Introduction
The main motivation of this paper is to construct generating functions for some certain new families of numbers and polynomials including generalized Apostol-Daehee numbers attached to Dirichlet character and the others In order construct generating functions functions for the generalized Apostol-type numbers and polynomials, we can use the p-adic q-Volkenborn integrals techniques including integral equations of the bosonic and the fermionic p-adic integrals on p-adic integers. The generalized Apostol-type numbers and polynomials are related to some well-known numbers and polynomials such as the generalized Bernoulli numbers and polynomials and the generalized Euler numbers and polynomials, the Apostol-Bernoulli numbers and polynomials and the Apostol-Euler numbers and polynomials, the Stirling numbers of the first and of the second kind, the Daehee numbers and polynomials and also the Changhee numbers and polynomials. The special functions and numbers have many applications in almost all branches of mathematics and the other science.
In this paper, we use the following notations, definitions and some families of the special numbers and polynomials.
Throughout this paper is a complex number. is a non-trivial Dirichlet character with conductor d.
Let r be a positive integer, and let ≠ 1 be any nontrivial r -th root of 1. Observe that the -Bernoulli numbers are reduced to the Apostol-Bernoulli numbers and the twisted Bernoulli numbers (cf. Kim et al., 2008, Theorem 1, p. 439; see also the references cited in each of these earlier works).
We (Simsek, 2016) gave the following functional equation: Using the above functional equation, we have (cf. Simsek, 2016) The generalized Apostol-Bernoulli numbers attached to Dirichlet character,  n, ( ) are defined by means of the following generating function: Apostol, 1951;Kim, 2007Kim, , 2008Kim & Son, 2007;Srivastava et al., 2005; see also the references cited in each of these earlier works).
The generalized Apostol-Euler numbers attached to Dirichlet character,  n, (x, ) are defined by means of the following generating function: Kim, 2007Kim, , 2008Srivastava et al., 2005; see also the references cited in each of these earlier works).
By combining (1.6) with (1.5), we have for the trivial character ≡ 1, we have (cf. Kim, 2007Kim, , 2008Srivastava et al., 2005) The Stirling numbers of the first kind, S 1 (n, k) are defined by means of the following generating function: Some properties of the Stirling numbers of the first kind are given as follows: For these numbers, recurrence relation is given by (cf. Charalambides, 2002;Roman, 2005;Simsek, 2013; and see also the references cited in each of these earlier works).
The Bernoulli polynomials of the second kind, b n (x) are defined by means of the following generating function (cf. Roman, 2005, pp. 113-117; see also the references cited in each of these earlier works): The Bernoulli numbers of the second kind b n (0) are defined by means of the following generating function: These numbers are computed by the following formula: where n,1 denotes the Kronecker delta (cf. Roman, 2005, p. 116). The Bernoulli polynomials of the second kind are defined by Substituting x = 0 into the above equation, one has The numbers b n (0) are also so-called the Cauchy numbers (cf. Charalambides, 2002;Qi, 2014;Roman, 2005, p. 116; see also the references cited in each of these earlier works). Kim et al. (2016) gave a computation method for the Bernoulli polynomials of the second kind are defined as follows: and also Roman (2005, p. 115) gave Using the above formula for the Bernoulli polynomials and numbers of the second kind, few of these numbers are computed as follows, respectively: and The generating function for the Stirling numbers of the second kind, S 2 (n, k) is given as follows: where k ∈ ℕ 0 . Some properties of these numbers are given as follows: S 2 (0, 0) = 1, S 2 (n, k) = 0 if k > n, S 2 (n, 0) = 0 if n > 0 and also recurrence relation is given by (cf. Charalambides, 2002;Roman, 2005;Srivastava & Choi, 2012; see also the references cited in each of these earlier works): The Peters polynomials s k (x; , ), which are Sheffer polynomials, are defined by means of the following generating functions (cf. Jordan, 1950;Roman, 2005): If = 1, then the polynomials s k (x; , ) are reduced to the Boole polynomials. If = 1 and = 1, then these polynomials are also reduced to the Changhee polynomials, which are given in Section 3 (cf. Kim & Kim, 2014;Roman, 2005). (1.10) S 2 (n, k) t n n! , S 2 (n + 1, k) = S 2 (n, k − 1) + kS 2 (n, k) In this section, we need the following definitions and notations.
Let ℤ p be a set of p-adic integers. Let be a field with a complete valuation and C 1 (ℤ p → ) be a set of continuous derivative functions. That is C 1 (ℤ p → ) is contained in the following set Definition 1 (Schikhof, 1984, p. 167, Definition 55.1) The Volkenborn integral (p-adic bosonic integral) of the function is given by We observe that 1 (x) = 1 x + p N ℤ p is the Haar distribution, defined by (cf. Hu & Kim, 2017;Kim, 2009, Schikhof, 1984; see also the references cited in each of these earlier works): In work of Kim (2002), the Volkenborn integral is also so-called the bosonic p-adic Volkenborn integral on ℤ p .
The Volkenborn integral in terms of the Mahler coefficients is given by the following formula: where (cf. Schikhof, 1984, p. 168, Proposition 55.3): From the above observation, we have the theorem: Theorem 1 Theorem 1 was proved by Schikhof (1984).
Let f :ℤ p → be an analytic function and f (x) = ∞ ∑ n=0 a n x n with x ∈ ℤ p . The Volkenborn integral of this analytic function is given by (cf. Schikhof, 1984, Proposition 55.4, p. 168): The following property is very important to our new results on special numbers: where (cf. Kim, 2002Kim, , 2006Schikhof, 1984; see also the references cited in each of these earlier works) The p-adic q-Volkenbo.rn integral was defined by Kim (2002). The distribution on ℤ p is given by where q ∈ ℂ p with | 1 − q | p < 1 and (cf. Kim, 2002) The p-adic q-integral of a function f ∈ C 1 (ℤ p → ) is defined by Kim (2002) as follows: Observe that The Witt's formula for the Bernoulli numbers and polynomials are given as follows, respectively and (cf. Kim, 2002Kim, , 2006Schikhof, 1984; see also the references cited in each of these earlier works) Let f ∈ C 1 (ℤ p → ) and Kim (2008, Theorem 1) defined the following functional equation for the q-bosonic p-adic Volkenborn integral on ℤ p as follows: where n is a positive integer. (1.13) lim q→1 x:q = x. (1.14) The fermionic p-adic integral on ℤ p is given by where (cf. Kim, 2009Kim, , 2006. Let p be a fixed prime. For a fixed positive integer d with (p, d) = 1, we set (see Kim, 2002) and a + dp N ℤ p = x ∈ | x ≡ a mod dp N where a ∈ ℤ satisfies the condition 0 ≤ a < dp N . Let f ∈ UD(ℤ p , ℂ p ). In work of Kim (2003), we see that The following integral equation is given by Kim (2008 We summarize our paper as follows: In Section 2, using the bosonic p-adic integral on ℤ p , we give generating functions for the generalized Apostol-Daehee numbers and polynomials attached to Dirichlet character . These numbers are related to many well-known numbers and polynomials. We also give not only relations between these numbers, the -Bernoulli numbers, the Stirling numbers, the Bernoulli numbers of the second kind, the generalized Bernoulli numbers, the generalized Euler numbers and the Daehee numbers and polynomials, but also a bosonic integral representation of these numbers. In Section 3, we give p-adic q-Volkenborn integral representation for the generalized Apostol-type Daehee numbers and polynomials with combinatorial sums.

Generalized Apostol-Daehee numbers attached to Dirichlet character on the bosonic p-adic integral
In this section, using the bosonic p-adic integral on ℤ p , we construct generating functions for the generalized Apostol-Daehee numbers and polynomials attached to Dirichlet character . We give relations between these numbers, the -Bernoulli numbers, the Stirling numbers, the Bernoulli numbers of the second kind, the generalized Bernoulli numbers, the generalized Euler numbers and the Daehee numbers and polynomials. We also give bosonic integral representation of these numbers. Firstly, we give some standard notations for the Volkenborn integral.
Let be a non-trivial Dirichlet character with conductor d. Let ∈ ℤ p . We set (1.20) Substituting (2.1) into (1.16), we get Using the above integral equation, we define the following generating function for generalized Apostol-Daehee numbers attached to Dirichlet character with conductor d as follows: and The generalized Apostol-Daehee polynomials attached to Dirichlet character are defined by means of the following generating function: so that, obviously, Combining the above function with (2.3), we have Therefore Comparing the coefficients of t n n! on both sides of the above equation, we arrive at the following theorem: Theorem 2 Let n ∈ ℕ 0 . Then we have (2.5) n, (z; , q) = n ∑ j=0 n j n−j (z) n−j j, ( , q).
In order to give a relation between the generalized Daehee numbers, the generalized Bernoulli polynomials, the Bernoulli numbers of the second kind and also the Stirling numbers of the first kind, using (2.3), we give the following functional equation: Combining the above functional equation with (1.1) and (1.9), we get Comparing the coefficients of t m m! on both sides of the above equation, we arrive at the following theorem: Theorem 3 Let m ∈ ℕ. Then we have If q → 1 in (2.6), we get the following corollary: