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The generating function approach is directly connected to the multipole expansion in electrostatics, as explained below, and is how the polynomials were first defined by Legendre in 1782. Definition via differential equation
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 ...
When in addition m is even, the function is a polynomial. When m is zero and ℓ integer, these functions are identical to the Legendre polynomials. 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.
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.
The main article gives examples of generating functions for many sequences. Other examples of generating function variants include Dirichlet generating functions (DGFs), Lambert series, and Newton series. In this article we focus on transformations of generating functions in mathematics and keep a running list of useful transformations and ...
One of the most well-known of these, the Laplace expansion for the three-variable Laplace equation, is given in terms of the generating function for Legendre polynomials, | ′ | = = < > + (), which has been written in terms of spherical coordinates (,,). The less than (greater than) notation means, take the primed or unprimed spherical ...
Here, the eigenvector, (), is a scaled and shifted discrete Chebyshev polynomial, and the corresponding eigenvalue, (), is the Legendre polynomial. Relation to Bessel functions [ edit ]
Several other classes of orthogonal polynomials, up to constant factors, are special cases of Jacobi polynomials, so these can be expressed using 2 F 1 as well. This includes Legendre polynomials and Chebyshev polynomials. A wide range of integrals of elementary functions can be expressed using the hypergeometric function, e.g.: