A Note on Legendre-Based Multi Poly-Eule Polynomials (original) (raw)

Summation Formulae for the Legendre Polynomials

Acta Mathematica Universitatis Comenianae

In this paper, summation formulae for the 2-variable Legendre poly- nomials in terms of certain multi-variable special polynomials are derived. Several summation formulae for the classical Legendre polynomials are also obtained as ap- plications. Further, Hermite-Legendre polynomials are introduced and summation formulae for these polynomials are also established.

A Family of Generalized Legendre-Based Apostol-Type Polynomials

Axioms, 2022

Numerous polynomials, their extensions, and variations have been thoroughly explored, owing to their potential applications in a wide variety of research fields. The purpose of this work is to provide a unified family of Legendre-based generalized Apostol-Bernoulli, Apostol-Euler, and Apostol-Genocchi polynomials, with appropriate constraints for the Maclaurin series. Then we look at the formulae and identities that are involved, including an integral formula, differential formulas, addition formulas, implicit summation formulas, and general symmetry identities. We also provide an explicit representation for these new polynomials. Due to the generality of the findings given here, various formulae and identities for relatively simple polynomials and numbers, such as generalized Bernoulli, Euler, and Genocchi numbers and polynomials, are indicated to be deducible. Furthermore, we employ the umbral calculus theory to offer some additional formulae for these new polynomials.

A new class of generalized Laguerre–Euler polynomials

Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, 2018

A variety of polynomials, their extensions and variants have been extensively investigated, due mainly to their potential of applications in diverse research areas. In this sequel, we introduce a new class of generalized Laguerre-Euler polynomials and present certain potentially useful formulas and identities such as implicit summation formulae and symmetry identities. The new class of polynomials introduced and the results presented here, being very general, are shown to be specialized to reduce to various known polynomials and to yield some known and new formulas and identities, respectively.

Certain generating funtion of generalized Apostol type Legendre-based polynomials

2018

In this paper, we aim to introduce a generating function for generalized Apostol type Legendre-Based polynomials which extends some known results. We also deduce some properties of the generalized Apostol-Bernoulli polynomials, the generalized Apostol-Euler polynomials and the generalized Apostol-Genocchi polynomials of higher order. By making use of the generating function method and some functional equations mentioned in the paper, we conduct a further investigation in order to obtain some implicit summation formulae and general symmetry identities for the generalized Apostol type Legendre-Based polynomials.

A note on multi Poly-Euler numbers and Bernoulli polynomials

In this paper we introduce the generalization of Multi Poly-Euler polynomials and we investigate some relationship involving Multi Poly-Euler polynomials. Obtaining a closed formula for generalization of Multi Poly-Euler numbers therefore seems to be a natural and important problem.

A New Class of Generalized Polynomials Associated with Hermite and Euler Polynomials

Mediterranean Journal of Mathematics, 2015

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.

New Generalization of Eulerian polynomials and Their Application

J. Ana. Num. Theor. 2, No. 2, 59-63 (2014)

In the present paper, we introduce Eulerian polynomials with a and b parameters and give the definition of them. By using the definition of generating function for our polynomials, we derive some new identities in Theory of Analytic Numbers. Also, we give relations between Eulerian polynomials with a and b parameters, Bernstein polynomials, Poly-logarithm function, Bernoulli numbers and Euler numbers. Moreover, we see that our polynomials at a = 1 are related to Euler-Zeta function at negative inetegers which we express in this paper.