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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 L_{n}(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 L_{n}(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)

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