Legendre polynomials

The Legendre polynomials, sometimes called Legendre functions of the first kind, are defined as solutions of Legendre's differential equation:

Solutions corresponding to the non-negative integer n can be expressed using Rodrigues' formula

or can be constructed using the three term recurrence relation:

P(x) = 1
P(x) = x
(n+1)Pn+1 (x) = (2n+1)x P(x)-nPn-1 (x)

Unit description

The recurrence relation given above is the most efficient way to calculate the Legendre polynomial. The LegendreCalculate subroutine uses this relation to calculate P(x) for any given x.

The LegendreSum subroutine calculates the sum of Legendre polynomials cP(x) + cP(x) + ... + cP(x) using Clenshaw's recurrence formula.

The LegendreCoefficients subroutine can represent P(x) as a sum of powers of x: c + cx + ... + cx n.

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