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:

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

The **LegendreSum** subroutine calculates the sum of Legendre polynomials *c _{0 }P_{0 }(x) + c_{1 }P_{1 }(x) + ... + c_{n }P_{n }(x)* using Clenshaw's recurrence formula.

The **LegendreCoefficients** subroutine can represent *P _{n }(x)* as a sum of powers of

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