Some Symmetric Identities for Degenerate Carlitz-type ( p , q ) -Euler Numbers and Polynomials

: In this paper we deﬁne the degenerate Carlitz-type ( p , q ) -Euler polynomials by generalizing the degenerate Euler numbers and polynomials, degenerate Carlitz-type q -Euler numbers and polynomials. We also give some theorems and exact formulas, which have a connection to degenerate Carlitz-type ( p , q ) -Euler numbers and polynomials.


Degenerate Carlitz-Type (p, q)-Euler Polynomials
In this section, we define the degenerate Carlitz-type (p, q)-Euler numbers and polynomials and make some of their properties.

Definition 2.
For 0 < q < p ≤ 1, the degenerate Carlitz-type (p, q)-Euler numbers E n,p,q (λ) and polynomials E n,p,q (x, λ) are related to the generating functions and respectively.
We see that By (5), it follows that By comparing the coefficients t n n! in the above equation, we have the following theorem.

Theorem 1.
For 0 < q < p ≤ 1 and n ∈ Z + , we have We make the degenerate Carlitz-type (p, q)-Euler number E n,p,q (λ). Some cases are We use t instead of Thus we have the following theorem. and Thus we have the below theorem from (9) and (10).

By (4) and (5), we get
Hence we have By comparing the coefficients of t m m! on both sides of (11), we have the following theorem.

Theorem 4.
For n ∈ Z + , we have We get that By (5) and (12), we get By comparing the coefficients of t n n! in the above equation, we have the theorem below.

Symmetric Properties about Degenerate Carlitz-Type (p, q)-Euler Numbers and Polynomials
In this section, we are going to get the main results of degenerate Carlitz-type (p, q)-Euler numbers and polynomials. We also make some symmetric identities for degenerate Carlitz-type (p, q)-Euler numbers and polynomials. Let w 1 and w 2 be odd positive integers. Remind that [xy] p,q = [x] p y ,q y [y] p,q for any x, y ∈ C.
By using w 1 x + w 1 i w 2 instead of x in Definition 2, use p by p w 2 , use q by q w 2 and use λ by respectively, we can get Since for any non-negative integer n and odd positive integer w 1 , there is the unique non-negative integer r such that n = w 1 r + j with 0 ≤ j ≤ w 1 − 1. So this can be written as We have the below formula using the above formula From a similar approach, we can have that Thus, we have the following theorem from (13) and (14).
Theorem 6. Let w 1 and w 2 be odd positive integers. Then one has Letting λ → 0 in Theorem 6, we can immediately obtain the symmetric identities for Carlitz-type (p, q)-Euler polynomials (see [10]) It follows that we show some special cases of Theorem 6. Let w 2 = 1 in Theorem 6, we have the multiplication theorem for the degenerate Carlitz-type (p, q)-Euler polynomials.
Let p = 1 in (15). This leads to the multiplication theorem about the degenerate Carlitz-type q-Euler polynomials Giving q → 1 in (16) induce to the multiplication theorem about the degenerate Euler polynomials E n (x, λ) = w n 1 If λ approaches to 0 in (17), this leads to the multiplication theorem about the Euler polynomials(see [15]) Let x = 0 in Theorem 6, then we have the following corollary. (−1) j q w 2 j E n,p w 1 ,q w 1 w 2 x + w 2 j w 1 , λ [w 1 ] p,q = n ∑ l=0 l ∑ k=0 l k S 1 (n, l)λ n−l p w 1 w 2 xk [2] q w 2 [w 2 ] k p,q [w 1 ] l−k p,q × E (k) l−k,p w 1 ,q w 1 (w 2 x)S l,k,p w 2 ,q w 2 (w 1 ).
Author Contributions: All authors contributed equally in writing this article. All authors read and approved the final manuscript.
Funding: This work was supported by the Dong-A university research fund.