Identities of Symmetry for Type 2 Bernoulli and Euler Polynomials

Abstract

The main purpose of this paper is to give several identities of symmetry for type 2 Bernoulli and Euler polynomials by considering certain quotients of bosonic p-adic and fermionic p-adic integrals on Z p , where p is an odd prime number. Indeed, they are symmetric identities involving type 2 Bernoulli polynomials and power sums of consecutive odd positive integers, and the ones involving type 2 Euler polynomials and alternating power sums of odd positive integers. Furthermore, we consider two random variables created from random variables having Laplace distributions and show their moments are given in terms of the type 2 Bernoulli and Euler numbers.

Some Identities on Type 2 Degenerate Bernoulli Polynomials of the Second Kind

“…As is known, the type 2 Bernoulli polynomials are defined by the generating function (1) t e t − e −t e xt = ∞ ∑ n=0 B * n (x) t n n! , (see [7]).…”

A Note on Type 2 Degenerate q-Euler Polynomials

“…Motivation for introducing the type 2 degenerate q-Euler polynomials and numbers is to study their number-theoretic and combinatorial properties, and their applications in mathematics, science and engineering. One novelty about this paper is that they arise naturally by means of the fermionic p-adic q-integrals so that it is possible to easily find some identities of symmetry for those polynomials and numbers, as it done, for example, in [1]. We spend the rest of this section in recalling what are needed in the sequel.…”

“…with the usual convention about replacing E * i by E * i . The Euler polynomials of degree n are given either by (see [1,[10][11][12][13][14][15][16][17][18][19])…”

“…where n ∈ N with n ≡ 1 (mod 2). The type 2 Euler numbers are the sequence E n , (n ≥ 0), of integers defined by (see [1])…”

A Note on the Degenerate Type of Complex Appell Polynomials

“…Further, there have been studies of various degenerate numbers and polynomials by means of degenerate types of generating functions, combinatorial methods, umbral calculus, and differential equations. For example, several authors have studied the degenerate types of Appell polynomials, such as Bernoulli and Euler polynomials (see [18][19][20][21][22][23]) and their complex version [24], degenerate gamma functions, degenerate Laplace transforms [25], and their modified ones [26].The research for degenerate versions of known special numbers and polynomials brought many valuable identities and properties into mathematics. In the future, we hope the results of the degenerate types of complex Appell polynomials can be further applicable to many different problems in various areas.The aim of this paper is to introduce Appell polynomials of a complex variable and their degenerate formulas and provide some of their properties and examples.…”