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Laguerre Polynomials. Laguerre differential equation. Recurrence formulas. Generating function. Associated Laguerre polynomials.

Laguerre differential equation. The equation

1)        xy" + (1- x)y' + ny = 0

If n = 0, 1, 2, 3, .... the solution of Laguerre’s equation is given by Laguerre polynomial Ln(x).

Laguerre polynomials. The polynomials given by Rodrigue’s formula

First few Laguerre polynomials

Generating function

Recurrence formulas

Orthogonality of Laguerre polynomials

Orthogonal series

where

Miscellaneous formulas

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Associated Laguerre polynomials

Laguerre’s associated differential equation. The equation

1)        xy" + (m + 1 - x)y' + (n - m )y = 0

For non-negative integers m and n, the solution of Laguerre’s associated equation is given by associated Laguerre polynomial .

Associated Laguerre polynomials. The polynomials given by

where Ln(x) are Laguerre polynomials.

The following hold:

First few associated Laguerre polynomials

Generating function for

Recurrence formulas

Orthogonality of associated Laguerre polynomials

Orthogonal series

where

Miscellaneous formulas

Source: Spiegel. Mathematical Handbook of Formulas and Tables. (Schaum)

References.

1. Spiegel. Mathematical Handbook of Formulas and Tables. (Schaum)