Some Identities on the Twisted q -Analogues of Catalan-Daehee Numbers and Polynomials

: In this paper, the author considers twisted q -analogues of Catalan-Daehee numbers and polynomials by using p -adic q -integral on Z p . We derive some explicit identities for those twisted numbers and polynomials related to various special numbers and polynomials.


Introduction
Let p be a fixed odd prime number. Throughout this paper, Z p , Q p and C p we denote the ring of p-adic integers, the field of p-adic rational numbers and the completion of algebraic closure of Q p . The p-adic norm | · | p is normally defined |p| p = 1 p . Let q be an indeterminate in C p with |1 − q| p < p − 1 p−1 . The q-analogue of x is defined by [x] q = 1 − q x 1 − q .
Note that lim Let f (x) be a uniformly differentiable function on Z p . Then the p-adic q-integral on Z p is defined by [1][2][3] From (1), we have where f (0) = d f (x) d x=0 . For n ∈ N, let T p be the p-adic locally constant space defined by where C p n = {w|w p n = 1} is the cyclic group of order p n .
For w ∈ T p , let us take f (x) = w x e xt . Then, by (1), we get Thus, by (3), we define the twisted q-Bernoulli numbers which are given by the generating function to be From (4), we note that with the usual convention about replacing B n q,w by B n,q,w . From (2) and (4), we have Thus, by (5), we get For |t| p < p − 1 p−1 , the twisted (λ, q)-Daehee polynomials are defined by generating function to be (cf. [4]) When x = 0, D n,q,w (λ) = D n,q,w (0|λ) are called the twisted (λ, q)-Daehee numbers. In particular, The twisted Catalan-Daehee numbers are defined by [5] If we take w = 1 in the twisted Catalan-Daehee numbers, d n = d n,1 , are the Catalan-Daehee numbers in [6][7][8].
We note that By replacing t by −4t in (9), we get where C m is the Catalan number.
From (8) and (10), Dolgy et al. showed a relation between the Catalan-Daehee numbers and the Catalan numbers in [6]; Catalan-Daehee numbers and polynomials were introduced in [7] and considered the family of linear differential equations arising from the generating function of those numbers in order to derive some explicit identities involving Catalan-Daehee numbers and Catalan numbers. In [8], several properties and identities associated with Catalan-Daehee numbers and polynomials were derived by utilizing umbral calculus techniques. Dolgy et al. gave some new identities for those numbers and polynomials derived from p-adic Volkenborn integrals on Z p in [6]. Recently, Ma et al. introduced and studied q-analogues of the Catalan-Daehee numbers and polynomials with the help of p-adic q-integral on Z p in [9]. The aim of this paper is to introduce q-analogues of the twisted Catalan-Daehee numbers and polynomials by using p-adic q-integral on Z p , and derive some explicit identities for those twisted numbers and polynomials related to various special numbers and polynomials.

The Twisted Q-Analogues of Catalan-Daehee Numbers
For t ∈ C p with |t| p < p In the view of (11), we define the twisted q-analogue of Catalan-Daehee numbers which are given by the generating function to be Note that lim q→1 d n,q,w = d n,w , (n ≥ 0), which is the twisted Catalan-Daehee numbers in [5].
From (7) and (12), we have Therefore, by comparing the coefficients on the both sides of (13), we obtain the following theorem.

By binomial expansion, we get
From (12) and (15), we obtain the following corollary. For the case w = 1 and q → 1, we have the following.
Let us take t = 1 4 (1 − e 2t ) in (12). Then we have On the other hand, where S 2 (n, k) (n, k ≥ 0) is the Stirling number of the second kind which is defined by Therefore, by (18) and (19), we obtain the following theorem.

Theorem 3.
For n ≥ 0, we have B n,q,w = n ∑ k=0 (−1) k S 2 (n, k)2 n−2k k!d k,q,w . Now, we observe that We define the twisted Catalan-Daehee polynomials which are given by the generating function to be When x = 0, d n,q,w = d n,q,w (0) (n ≥ 0) are the twisted Catalan-Daehee numbers in (12). Note that Thus, by (12), (20) and (21), we get By comparing the coefficients on the both sides (22), we obtain the following theorem. For the case w = 1 and q → 1, we have the following.

Conclusions
To summarize, we introduced twisted q-analogues of Catalan-Daehee numbers and polynomials and obtained several explicit expressions and identities related to them. We expressed the twisted q-analogues of Catalan-Daehee numbers in terms of the twisted (λ, q)-Daehee numbers, and of the twisted q-Bernoulli numbers and Stirling numbers of the first kind in Theorems 1 and 2. We also derived an identity involving the twisted q-Bernoulli numbers, twisted q-analogues of Catalan-Daehee numbers and Stirling numbers of the second kind in Theorem 3. In addition, we obtain an explicit expression for the twisted q-analogues of Catalan-Daehee polynomials which involve the twisted q-analogues of Catalan-Daehee numbers and Stirling numbers of the first kind in Theorem 4.
In recent years, many special numbers and polynomials have been studied by employing various methods, including: generating functions, p-adic analysis, combinatorial methods, umbral calculus, differential equations, probability theory and analytic number theory. We are now interested in continuing our research into the application of 'twisted' and 'q-analogue' versions of certain interesting special polynomials and numbers in the fields of physics, science, and engineering as well as mathematics.