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.
Highlights
Introduction and preliminariesThe two variable Laguerre polynomials Ln(x, y) are defined by the following generating function: 1 − yt exp ( −xt 1 − yt ) = ∑ ∞Ln(x, y) tn (| yt |< 1). (1) n=0
We find that the polynomials Bn(α)(x, ν) in (9) are related to Bernoulli polynomials and Hermite polynomials
We find the ordinary Hermite polynomials Hn(x) = Hn(2x, −1)
Summary
The two variable Laguerre polynomials Ln(x, y) are defined by the following generating function The polynomials Ln(x, y) are given by the following generating function (see [14]): eyt. Setting h(t, ν) = h(t, y) = yt in (8), we have the generalized Hermite–Bernoulli polynomials of two variables H Bn(α)(x, y) , which were introduced and investigated by Pathan [29], given by. Guo and Qi [17] (see [32]) introduced the following generalized Bernoulli numbers Bn(a, b) defined by tn at − bt := Bn(a, b) n!. Luo et al [27] generalized the numbers Bn(a, b) in (16) to introduce and investigate the generalized Bernoulli polynomials Bn(x; a, b, e) defined by t ext at − bt. We aim to introduce generalized Laguerre–Bernoulli polynomials LBn(α)(x, y, z; a, b, e) in (20) and investigate some of their properties such as explicit summation formulas, addition formulas, implicit formula, and symmetry identities
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