A novel theory of Legendre polynomials

Turkish Journal of Analysis and Number Theory, 2013, Vol. 1, No. 1, 1-3

In the present paper, we deal mainly with arithmetic properties of Legendre polynomials by using their orthogonality property. We show that Legendre polynomials are proportional with Bernoulli, Euler, Hermite and Bernstein polynomials.

We use the multivariable Hermite polynomials to derive integral representations of Chebyshev and Gegenbauer polynomials. It is shown that most of the properties of these classes of polynomials can be deduced in a fairly straightforward way from this representation, which proves a unifying framework for a large body of polynomial families, including forms of the Humbert and Bessel type, which are a natural consequence of the point of view developed in this paper.

Mathematical and Computer Modelling, 2012

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.

Journal of Physics: Conference Series, 2019

Legendre polynomials are obtained through well-known linear algebra methods based on Sturm-Liouville theory. A matrix corresponding to the Legendre differential operator is found and its eigenvalues are obtained. The elements of the eigenvectors obtained correspond to the Legendre polynomials. This method contrast in simplicity with standard methods based on solving Legendre differential equation by power series, using the Legendre generating function, using the Rodriguez formula for Legendre polynomials, or by a contour integral.

Mathematical and Computer Modelling, 2008

We combine the Lie algebraic methods and the technicalities associated with the monomialty principle to obtain new results concerning Legendre polynomial expansions.

Kragujevac Journal of Mathematics, 2020

The special polynomials of more than one variable provide new means of analysis for the solutions of a wide class of partial differential equations often encountered in physical problems. Motivated by their importance and potential for applications in a variety of research fields, recently, numerous polynomials and their extensions have been introduced and investigated. In this paper, we introduce a new family of Laguerre-based generalized Hermite-Euler polynomials, which are related to the Hermite, Laguerre and Euler polynomials and numbers. The results presented in this paper are based upon the theory of the generating functions. We derive summation formulas and related bilateral series associated with the newly introduced generating function. We also point out that the results presented here, being very general, can be specialized to give many known and new identities and formulas involving relatively simple numbers and polynomials.

Journal of Approximation Theory, 2012

In 1951, F. Brafman derived several "unusual" generating functions of classical orthogonal polynomials, in particular, of Legendre polynomials P n (x). His result was a consequence of Bailey's identity for a special case of Appell's hypergeometric function of the fourth type. In this paper, we present a generalization of Bailey's identity and its implication to generating functions of Legendre polynomials of the form ∞ n=0 u n P n (x)z n , where u n is an Apéry-like sequence, that is, a sequence satisfying (n + 1) 2 u n+1 = (an 2 + an + b)u n − cn 2 u n−1 where n ≥ 0 and u −1 = 0, u 0 = 1. Using both Brafman's generating functions and our results, we also give generating functions for rarefied Legendre polynomials and construct a new family of identities for 1/π.

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.

Notes on Number Theory and Discrete Mathematics, 2019

In the past years, many researchers have worked on degenerate polynomials and a variety of its extentions and variants can be found in literature. Following up, in this article, we firstly introduce the partially degenerate Legendre-Genocchi polynomials, and further define a new generalization of degenerate Legendre-Genocchi polynomials. From our generalization, we establish some implicit summation formulae and symmetry identities by the generating function of partially degenerate Legendre-Genocchi polynomials. Eventually, it can be found that some recently demonstrated summations and identities stated in the article, are special cases of our results.