Some Symmetry Identities for Carlitz’s Type Degenerate Twisted ( p , q ) -Euler Polynomials Related to Alternating Twisted ( p , q ) -Sums

: In this paper, we deﬁne a new form of Carlitz’s type degenerate twisted ( p , q ) -Euler numbers and polynomials by generalizing the degenerate Euler numbers and polynomials, Carlitz’s type degenerate q -Euler numbers and polynomials. Some interesting identities, explicit formulas, symmetric properties, a connection with Carlitz’s type degenerate twisted ( p , q ) -Euler numbers and polynomials are obtained. Finally, we investigate the zeros of the Carlitz’s type degenerate twisted ( p , q ) -Euler

In this paper we define a new form of Carlitz's type degenerate twisted (p, q)-Euler numbers and polynomials and study some theories of the Carlitz's type degenerate twisted (p, q)-Euler numbers and polynomials.
Throughout this paper, we always make use of the following classical notations: N denotes the set of natural numbers, Z denotes the set of integers, Z 0 = N ∪ {0} denotes the set of nonnegative integers, R denotes the set of real numbers, and C denotes the set of complex numbers.
For z ∈ C, the (p, q)-number is defined by The (p, q)-number is a natural generalization of the q-number, that is By using (p, q)-number, we define a new form of Carlitz's type degenerate twisted (p, q)-Euler numbers and polynomials, which generalized the previously known numbers and polynomials, including the degenerate Euler numbers and polynomials, Carlitz's type degenerate q-Euler numbers and polynomials (see [2,3,6,9]). We begin by recalling here the Carlitz's type twisted (p, q)-Euler numbers and polynomials (see [13]). Let ζ be rth root of 1 and ζ = 1 (see [13,14,16] and their values at z = 0 are called the Carlitz's type twisted (p, q)-Euler numbers and denoted E n,p,q,ζ .
Here is a brief introduction to the history for the reader. The following diagram shows the variations of the different types of Euler polynomials, degenerate Euler polynomials, (p, q)-Euler polynomials, degenerate (p, q)-Euler polynomials, and twisted (p, q)-Euler polynomials. Those polynomials in the first row and the second row of the diagram are studied by Carlitz [1,2], Cenkci, and Howard [3][4][5], Young [6], Hwang and Ryoo [9], Ryoo [10,11], Simsek [16], and Srivastava [17], respectively. The motivation of this paper is to investigate some interesting symmetric identities and explicit identities for Carlitz's type degenerate twisted (p, q)-Euler polynomials in the third row of the diagram.
In the following section, we define a new form of Carlitz's type degenerate twisted (p, q)-Euler numbers E n,p,q,ζ (µ) and polynomials E n,p,q,ζ (z, µ). After that we will investigate some their properties. In Section 2, Carlitz's type degenerate twisted (p, q)-Euler numbers E n,p,q,ζ (µ) and polynomials E n,p,,ζ (z, µ) are defined. We derive some of their relevant properties. In Section 3, first, we derive the symmetric properties for Carlitz's type degenerate twisted (p, q)-Euler numbers E n,p,q,ζ (µ) and polynomials E n,p,q,ζ (z, µ). Finally, we investigate the zeros of the Carlitz's type degenerate twisted (p, q)-Euler numbers E n,p,q,ζ (µ) and polynomials E n,p,q,ζ (z, µ) by using computer.
Putting p = 1 and ζ = 1, we have Proof. Since we have By comparing the coefficients t n n! in the above equation, the proof is completed.
Proof. By replacing t by Thus, by (5), the proof is completed.
Proof. By (1) and (2), we get Hence, by (8), we also have By comparing the coefficients t m m! on both sides of (9), the proof is completed.

Symmetric Properties about Carlitz's Type Degenerate Twisted (p, q)-Euler Numbers and Polynomials
In this section, we are going to obtain the main results of Carlitz's type degenerate twisted (p, q)-Euler numbers E n,p,q,ζ (µ) and polynomials E n,p,q,ζ (z, µ). We also establish some interesting symmetric identities for Carlitz's type degenerate twisted (p, q)-Euler numbers E n,p,q,ζ (µ) and polynomials E n,p,q,ζ (z, µ).

Theorem 6. Let a and b be odd positive integers. Then one has
Proof. Observe that [zy] p,q = [z] p y ,q y [y] p,q for any z, y ∈ C. By substitute az + ai b for z in Definition 2, replace p by p b , replace q by q b , and replace ζ by ζ b , respectively, we derive Since for any non-negative integer n and odd positive integer a, there exist unique non-negative integer r such that n = ar + j with 0 ≤ j ≤ a − 1. So this can be written as [abz + ai + abr + bj] p,q µ .
It follows from the above equation that From the similar method, we can obtain that Thus, from (11) and (12), the proof is completed.
It follows that we show some special cases of Theorem 6. Setting b = 1 in Theorem 6, we have the multiplication theorem for the Carlitz's type degenerate twisted (p, q)-Euler polynomials E n,p,q,ζ (z, µ).

Corollary 1. Let a be odd positive integer. Then one has
Let z = 0 in Theorem 6, we have the following corollary.

Corollary 2. Let a and b be odd positive integers. Then it has
By Theorem 3 and Corollary 2, we have the below theorem.
Theorem 7. Let a and b be odd positive integers. Then We get another result by applying the addition theorem about the Carlitz-type twisted (h, p, q)-Euler polynomials E (h) n,p,q,ζ (x) (see [14]).

Theorem 8. Let a and b be odd positive integers. Then we have
where S l,k,p,q,ζ (a) = ∑ a−1 i=0 (−1) i q (l−k+1)i ζ i [i] k p,q is called as the alternating twisted (p, q)-sums.
Proof. From (3), Theorem 3, and Theorem 6, we have Therefore, we induce that and l−k,p a ,q a ,ζ a (bz)S l,k,p b ,q b ,ζ b (a).
In particular, the case a = 3 in Corollary 1 gives the triplication formula for Carlitz's type degenerate twisted (p, q)-Euler polynomials Setting p = 1 in (13) and (16) leads to the familiar multiplication theorem for the Carlitz's type degenerate twisted q-Euler polynomials and the triplication formula for Carlitz's type degenerate twisted q-Euler polynomials Letting q → 1 in (17) and (18) leads to the familiar multiplication theorem for the degenerate twisted Euler polynomials E n,ζ (z, µ) = a n a−1 Our numerical results for approximate solutions of complex zeros of E n,p,q,ζ (z, µ) = 0 are displayed (Tables 1 and 2). The * mark in Table 1 means that there is no solution of E n,p,q,ζ (z, µ) = 0. As a result of numerical experiments, it was found that ther is no solution of E n,p,q,ζ (z, µ) = 0 for p = 1/2, q = 1/10, n = 2 k , k = 1, 2, 3, . . . We hope to verify that there is no solution of E n,p,q,ζ (z, µ) = 0 for p = 1/2, q = 1/10, n = 2 k , k = 1, 2, 3, . . . We can see a regular pattern of the roots of the E n,p,q,ζ (z, µ) = 0 and also hope to verify the same kind of regular structure of the roots of the E n,p,q,ζ (z, µ) = 0 (Table 1).
We investigate the zeros of the E n,p,q,ζ (z, µ) = 0 by using a computer. We plot the zeros of the E n,p,q,ζ (z, µ) = 0 for z ∈ C (Figure 1).

Conclusions and Discussion
In our previous paper [13], we studied some identities of symmetry on the Carlitztype twisted (p, q)-Euler numbers and polynomials. The motivation of this paper is to construct the Carlitz's type degenerate twisted (p, q)-Euler numbers and polynomials. We also investigate some explicit identities for the Carlitz's type degenerate twisted (p, q)-Euler polynomials in the third row of the diagram at page 3. Therefore, we construct the