Efficient Legendre polynomials transforms: from recurrence relations to Schoenberg's theorem (original) (raw)

Generating functions of Legendre polynomials: A tribute to Fred Brafman

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.

A novel theory of Legendre polynomials

Mathematical and Computer Modelling, 2011

We reformulate the theory of Legendre polynomials using the method of integral transforms, which allow us to express them in terms of Hermite polynomials. We show that this allows a self consistent point of view to their relevant properties and the possibility of framing generalized forms like the Humbert polynomials within the same framework. The multi-index multi-variable case is touched on.

Simple Approach to Special Polynomials: Laguerre, Hermite, Legendre, Tchebycheff, and Gegenbauer

Applied Mathematics [Working Title]

Special polynomials: Laguerre, Hermite, Legendre, Tchebycheff and Gegenbauer are obtained through well-known linear algebra methods based on Sturm-Liouville theory. A matrix corresponding to the differential operator is found and its eigenvalues are obtained. The elements of the eigenvectors obtained correspond to each mentioned polynomial. This method contrasts in simplicity with standard methods based on solving the differential equation by means of power series, obtaining them through a generating function, using the Rodrigues formula for each polynomial, or by means of a contour integral.

Recurrences and Legendre Transform

1992

A binomial identity ((1) below), which relates the famous Apéry numbers and the sums of cubes of binomial coefficients (for which Franel has established a recurrence relation almost 100 years ago), can be seen as a particular instance of a Legendre transform between sequences. A proof of this identity can be based on the more general fact that the Apéry and Franel recurrence relations themselves are conjugate via Legendre transform. This motivates a closer look at conjugacy of sequences satisfying linear recurrence relations with polynomial coefficients. The rôle of computer-aided proof and verification in the study of binomial identities and recurrence relations is illustrated, and potential applications of conjugacy in diophantine approximation are mentioned. This article is an expanded version of a talk given at the 29. meeting of the Séminaire Lothringien de Combinatoire, Thurnau, september 1992.

Legendre Polynomials: a Simple Methodology

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.

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 new and efficient method for the computation of Legendre coefficients

An efficient procedure for the computation of the coefficients of Legendre expansions is here presented. We prove that the Legendre coefficients associated with a function f(x) can be represented as the Fourier coefficients of an Abel-type transform of f(x). The computation of N Legendre coefficients can then be performed in O(N log N) operations with a single Fast Fourier Transform of the Abel-type transform of f(x).

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.