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:
P0(x) = 1The recurrence relation given above is the most efficient way to calculate the Legendre polynomial. The LegendreCalculate subroutine uses this relation to calculate Pn(x) for any given x.
The LegendreSum subroutine calculates the sum of Legendre polynomials c0P0(x) + c1P1(x) + ... + cnPn(x) using Clenshaw's recurrence formula.
The LegendreCoefficients subroutine can represent Pn(x) as a sum of powers of x: c0 + c1x + ... + cnxn.
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