REFERENCE EQUATIONS 3 (Legendre Functions, Identities and Integrals (original) (raw)
https://doi.org/10.13140/RG.2.2.32477.41440
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Abstract
Equations involving legendre functions are shown
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On some dual integral equations involving Legendre and associated Legendre functions
Journal of the Indian Institute of Science, 1993
Dual integral equations involving Legendre and associated Legendre functions as kernels are considered in this paper. Except for one pair, these dual integral equations are first reduced to solving some appropriate ordinary differential eauations. Invoking. the inversion formulae for Abel internal eauations. closed-fonn solutions to -. these dual integral equations are obtained in most cases and in other cases they are reduced to some appropriate Fredholm integral equation of the second kind. For the exceptional dual integral equations pair which involves the associated Legendre function as kernel, a direct method of the use of Abel integral equation is applied to obtain the closed-form solution. As an example of application of these dual integral equations a problem arising from mathematical physics is considered.
Legendre’s equation is key in various branches of physics. Its general solution is a linear function space, spanned by the Legendre functions of first and second kind. In physics however, commonly the only acceptable members of this set are the Legendre polynomials. Quantization of the eigenvalues of Legendre’s operator is a consequence of this. We present and explain a stand-alone, in-depth argument for rejecting all solutions of Legendre’s equation, but the polynomial ones, in physics. We show that the combination of the linearity, the mirror symmetry and the signature of the regular singular points of Legendre’s equation is quintessential to the argument. We demonstrate that the evenness or oddness of the Legendre polynomials is a consequence of the same ingredients.
Generating functions of Legendre polynomials: A tribute to Fred Brafman
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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.
Expansion of a class of functions into an integral involving associated Legendre functions
International Journal of Mathematics and Mathematical Sciences, 1994
A theorem for expansion of a class of functions into an integral involving associated Legendre functions is obtained in this paper. This is a somewhat general integral expansion formula for a functionf(x)defined in(x1,x2)where-1<x1<x2<1, which is perhaps useful in solving certain boundary value problems of mathematical physics and of elasticity involving conical boundaries.
The Study of Triple Integral Equations with Generalized Legendre Functions
Abstract and Applied Analysis, 2008
A method is developed for solutions of two sets of triple integral equations involving associated Legendre functions of imaginary arguments. The solution of each set of triple integral equations involving associated Legendre functions is reduced to a Fredholm integral equation of the second kind which can be solved numerically.
Some Identities Involving the Legendre's Chi-Function
Communications of the Korean Mathematical Society, 2007
Since the time of Euler, the dilogarithm and polylogarithm functions have been studied by many mathematicians who used various notations for the dilogarithm function Li 2 (z). These functions are related to many other mathematical functions and have a variety of application. The main objective of this paper is to present corrected versions of two equivalent factorization formulas involving the Legendre's Chi-function χ 2 and an evaluation of a class of integrals which is useful to evaluate some integrals associated with the dilogarithm function.
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
On Legendre functions of imaginary degree and associated integral transforms
Journal of Mathematical Analysis and Applications, 2003
New integral representations, asymptotic formulas, and series expansions in powers of tanh(t/2) are obtained for the imaginary and real parts of the Legendre function P iξ (cosh t). Coefficients of these series expansions are orthogonal polynomials in the real variable ξ . A number of relations for these orthogonal polynomials are obtained on the basis of the generating function. Several inversion theorems are proven for the integral transforms involving the Legendre function of imaginary degree. In many cases it is preferable to employ these transforms, than Mehler-Fok transforms, since conditions placed on functions are less restrictive.
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A triangular property of the associated Legendre functions
Journal of Mathematical Physics, 1990
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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.
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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.
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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.
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