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The differential equation admits another, non-polynomial solution, the Legendre functions of the second kind. A two-parameter generalization of (Eq. A two-parameter generalization of (Eq. 1 ) is called Legendre's general differential equation, solved by the Associated Legendre polynomials .
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, the associated Legendre polynomials are the canonical solutions of the general Legendre equation () + [(+)] =,or equivalently [() ()] + [(+)] =,where the indices ℓ and m (which are integers) are referred to as the degree and order of the associated Legendre polynomial respectively.
Similar formulae hold for many other sequences of orthogonal functions arising from Sturm–Liouville equations, and these are also called the Rodrigues formula (or Rodrigues' type formula) for that case, especially when the resulting sequence is polynomial.
Furthermore, a change of variables t = cos θ transforms this equation into the Legendre equation, whose solution is a multiple of the associated Legendre polynomial P m ℓ (cos θ). Finally, the equation for R has solutions of the form R(r) = A r ℓ + B r −ℓ − 1; requiring the solution to be regular throughout R 3 forces B = 0. [3]
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.
The differential equation is said to be in Sturm–Liouville form or self-adjoint form.All second-order linear homogenous ordinary differential equations can be recast in the form on the left-hand side of by multiplying both sides of the equation by an appropriate integrating factor (although the same is not true of second-order partial differential equations, or if y is a vector).
In this case p 1 (x) = 1/x has a pole of first order at x = 0. When α ≠ 0, p 0 (x) = (1 − α 2 /x 2) has a pole of second order at x = 0. Thus this equation has a regular singularity at 0. To see what happens when x → ∞ one has to use a Möbius transformation, for example = /.