Abstract
A variety of polynomials, their extensions, and variants, have been extensively investigated, mainly due to their potential applications in diverse research areas. Motivated by their importance and potential for applications in a variety of research fields, numerous polynomials and their extensions have recently been introduced and investigated. In this paper, we introduce generalized Laguerre poly-Genocchi polynomials and investigate some of their properties and identities, which were found to extend some known results. Among them, an implicit summation formula and addition-symmetry identities for generalized Laguerre poly-Genocchi polynomials are derived. The results presented here, being very general, are pointed out to reduce to yield formulas and identities for relatively simple polynomials and numbers.
Highlights
The generating function of the two variable Laguerre polynomials L p (u, v) [1] is defined by ∞ ζp∑ L p (u, v) p!, evζ C0 = (1)p =0 where C0 (u) is the 0-th order Tricomi function [2] C0 (u) =(−1)s us s=0 ( s! ) ∑ (2)we find from Equation (1) that p
One of the important classes of polynomial sequences is the class of Appell polynomial sequences
The Appell polynomial sequences are defined by the following generating function: A(u, ζ ) = A(ζ )euζ =
Summary
The generating function of the two variable Laguerre polynomials L p (u, v) [1] is defined by. Kaneko [3] introduced and investigated generalized poly-Bernoulli numbers by means of the generating functions p. In References [4,5,6,7], Jolany et al introduced and studied the generalized poly-Bernoulli numbers and polynomials, which appear in the following power series. The organization of this paper is given as follows: In Section 2, we introduce generalized Laguerre poly-Genocchi polynomials L G p (u, v, w) and develop elementary properties by using generating functions for the numbers. We introduce and investigate the generalized Laguerre poly-Genocchi polynomials as follows: 2Lik (1 − e−ζ ) vζ +wζ 2 e. Setting u = 0 in (20), we obtain the Hermite poly-Genocchi polynomials by Khan [12], defined as eζ + 1.
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