A new class of generalized polynomials associated with Laguerre and Bernoulli polynomials

Abstract

Motivated by their importance and potential for applications in certain problems in number theory, combinatorics, classical and numerical analysis, and other fields of applied mathematics, a variety of polynomials and numbers with their variants and extensions have recently been introduced and investigated. In this paper, we aim to introduce generalized Laguerre-Bernoulli polynomials and investigate some of their properties such as explicit summation formulas, addition formulas, implicit formulas, and symmetry identities. Relevant connections of the results presented here with those relatively simple numbers and polynomials are considered.

A Study of Generalized Laguerre Poly-Genocchi Polynomials

“…Furthermore, the generalized Laguerre poly-Genocchi polynomials L G (k) p (u, v, w) in Equation 20, being very general, can be specialized to yield various known polynomials and numbers, for example, Genocchi numbers G n , Genocchi polynomials G n (x), and Hermite-Genocchi polynomials H G n (u, v) (see Reference [12]). In this regard, the results presented here can be specialized to yield or be closely connected with some known identities and formulas (see, e.g., References [14][15][16][17][18][19][20][21]23,25]), and the references cited therein). Therefore, the results presented in this article could potentially be useful in problems that arise in the aforementioned fields.…”

“…Symmetry identities involving various polynomials have been presented (e.g., References [10,[14][15][16][17][18][20][21][22]). As in the above-cited works, here we establish some addition-symmetry identities involving generalized Laguerre poly-Genocchi polynomials L G…”

A new class of generalized polynomials involving Laguerre and Euler polynomials

“…The two variable Laguerre polynomials L n (x, y) are generated by (see [8,18])…”

“…Also, equivalently, the polynomials L n (x, y) are given by (see [9,18])…”

The r-central factorial numbers with even indices

“…In recent years, many mathematicians introduced and studied various degenerate and extended versions of a lot of old and new special numbers and polynomials, namely Bernoulli numbers and polynomials, Eulerian numbers and polynomials, Daehee numbers, Bell polynomials, and type 2 Bernoulli polynomials of the second kind, to name a few (see [1,2,11,15,17,28] and the references therein). Here, we are interested in extended versions of the central factorial numbers.…”