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In general, when ℓ and m are integers, the regular solutions are sometimes called "associated Legendre polynomials", even though they are not polynomials when m is odd. The fully general class of functions with arbitrary real or complex values of ℓ and m are Legendre functions .
The general Legendre equation reads ″ ′ + [(+)] =, where the numbers λ and μ may be complex, and are called the degree and order of the relevant function, respectively. . The polynomial solutions when λ is an integer (denoted n), and μ = 0 are the Legendre polynomials P n; and when λ is an integer (denoted n), and μ = m is also an integer with | m | < n are the associated Legendre ...
In mathematics, Legendre polynomials, named after Adrien-Marie Legendre (1782), are a system of complete and orthogonal polynomials with a wide number of mathematical properties and numerous applications. They can be defined in many ways, and the various definitions highlight different aspects as well as suggest generalizations and connections ...
Classical orthogonal polynomials appeared in the early 19th century in the works of Adrien-Marie Legendre, who introduced the Legendre polynomials. In the late 19th century, the study of continued fractions to solve the moment problem by P. L. Chebyshev and then A.A. Markov and T.J. Stieltjes led to the general notion of orthogonal polynomials.
1.1 Finding associated Legendre polynomials. 2 comments. 1.2 Applying the chain rule more than once. 5 comments. 1.3 perfect number. 5 comments. 1.4 Binomial related ...
Let (()) = be a sequence of orthogonal polynomials defined on the interval [,] satisfying the orthogonality condition () =,, where () is a suitable weight function, is a constant depending on , and , is the Kronecker delta.
Associated Legendre polynomials; Angular momentum; Angular momentum coupling; Total angular momentum quantum number; Azimuthal quantum number; Table of Clebsch–Gordan coefficients; Wigner D-matrix; Wigner–Eckart theorem; Angular momentum diagrams (quantum mechanics) Clebsch–Gordan coefficient for SU(3) Littlewood–Richardson coefficient
He is remembered for the Gegenbauer polynomials, a class of orthogonal polynomials. They are obtained from the hypergeometric series in certain cases where the series is in fact finite. The Gegenbauer polynomials are solutions to the Gegenbauer differential equation and are generalizations of the associated Legendre polynomials.