Laguerre polynomials

Laguerre polynomials are defined as solutions of Laguerre's differential equation:

xy'' + (1-x)y' + ny = 0

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

or can be constructed using the three term recurrence relation:

L₀ (x) = 1
L₁ (x) = 1-x
(n+1)L_n+1 (x) = (2n+1-x)L_n (x)-nL_n-1 (x)

Unit description

The recurrence relation given above is the most efficient way to calculate the Laguerre polynomial. The LaguerreCalculate subroutine uses this relation to calculate L_n (x) for any given x.

The LaguerreSum subroutine calculates the sum of Laguerre polynomials c₀ L₀ (x) + c₁ L₁ (x) + ... + c_n L_n (x) using Clenshaw's recurrence formula.

The LaguerreCoefficients subroutine can represent L_n (x) as a sum of powers of x: c₀  + c₁ x + ... + c_n x ⁿ.

This article is intended for personal use only.

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