A triangular property of the associated Legendre functions (original) (raw)

A Combinatorial Formula for the Associated Legendre Functions of Integer Degree

Advances in Mathematics, 2000

We apply inverse scattering theory to a Schrödinger operator with a regular reflectionless Pöschl-Teller potential on the line, to arrive at a combinatorial formula for the associated Legendre functions of integer degree. The expansion coefficients in the combinatorial formula are identified as dimensions of irreducible representations of gl(N), where N corresponds to the degree of the associated Legendre function. As an application, combinatorial formulas for the zonal spherical functions on the real hyperboloids H 2N +3,1 = SO 0 (2N + 3, 1; R)/SO 0 (2N + 2, 1; R), H 1,2N +3 + = SO 0 (2N + 3, 1; R)/SO(2N + 3; R) and the sphere S 2N +3 = SO(2N + 4; R)/SO(2N + 3; R) are presented. Dedicated to Professor Richard A. Askey on the occasion of his 65th birthday. Résumé En appliquant la théorie de la diffusion inverseà un opérateur de Schrödinger avec un potentiel de Pöschl-Teller régulier et sans réflexion on arriveà une formule combinatoire pour les fonctions de Legendre associées de degré entier. Les coefficients dans cette formule ne sont que les dimensions de représentations irréductibles de gl(N), où N correspond au degré de la fonction de Legendre associée. Comme exemples, on calcule des fonctions zonales sur les hyperboloïdes réels H 2N +3,1 = SO 0

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.

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.

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.