Some Symmetric Identities for the Multiple (p, q)-Hurwitz-Euler eta Function

Abstract

The main purpose of this paper is to find some interesting symmetric identities for the ( p , q ) -Hurwitz-Euler eta function in a complex field. Firstly, we define the multiple ( p , q ) -Hurwitz-Euler eta function by generalizing the Carlitz’s form ( p , q ) -Euler numbers and polynomials. We find some formulas and properties involved in Carlitz’s form ( p , q ) -Euler numbers and polynomials with higher order. We find new symmetric identities for multiple ( p , q ) -Hurwitz-Euler eta functions. We also obtain symmetric identities for Carlitz’s form ( p , q ) -Euler numbers and polynomials with higher order by using symmetry about multiple ( p , q ) -Hurwitz-Euler eta functions. Finally, we study the distribution and symmetric properties of the zero of Carlitz’s form ( p , q ) -Euler numbers and polynomials with higher order.

Symmetric Identities for Carlitz-Type Higher-Order Degenerate (p,q)-Euler Numbers and Polynomials

“…We also obtained the explicit identities related to the Carlitz-type higher-order (p, q)-Euler polynomials, the alternating (p, q)-sums of powers, and Stirling numbers (see Theorem 10 and Corollary 4). In particular, these results generalized some well-known properties relating degenerate Euler numbers and polynomials, degenerate Stirling numbers, alternating sums of powers, multiplication theorem, distribution relation, falling factorial, symmetry properties of the degenerate Euler numbers and polynomials (see [7][8][9][10][11][12][13][14][15][16][17][18]). In addition, in this paper, if we take r = 1, then [4] is the special case of this paper.…”

“…Taking λ = 0 in (11), we get the multiplication theorem for Carlitz-type high order (p, q)-Euler polynomials (see [11]). Corollary 1.…”

“…respectively (see [4]). Hwang and Ryoo [11] discussed some properties for Carlitz-type higher-order (p, q)-Euler numbers and polynomials. For r ∈ N and 0 < q < p ≤ 1, the Carlitz-type higher-order (p, q)-Euler polynomials E (1)…”

“…The following diagram shows the variations of the different types of degenerate Euler polynomials and Euler polynomials. Those polynomials in the first row and the third row of the diagram are studied by Hwang and Ryoo [4,11], Carlitz [7], Cenkci and Howard [9], Wu and Pan [12], Luo [13], and Srivastava [14], respectively. The study of these has produced beneficial results in combinatorics and number theory (see [4,7,9,[12][13][14][15][16][17][18]).…”

Symmetric Identities for Carlitz’s Type Higher-Order (p,q)-Genocchi Polynomials

“…A number of researchers have been looking into Bernoulli polynomials, Euler polynomials, and Genocchi polynomials (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]). This paper reviews Bernoulli polynomials, Euler polynomials, and Genocchi polynomials.…”