Website owner: James Miller
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)
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