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The LaguerreL function computes the nth Laguerre polynomial.
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If the first parameter is a non-negative integer, the LaguerreL function computes the nth generalized Laguerre polynomial with parameter a evaluated at x.
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If a is not specified, LaguerreL(n, x) computes the nth Laguerre polynomial which is equal to LaguerreL(n, 0, x).
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The generalized Laguerre polynomials are orthogonal on the interval with respect to the weight function . They satisfy:
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For positive integer a, the relationship for LaguerreL(n, a, x) and LaguerreL(n, x) is the following.
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Some references define the generalized Laguerre polynomials differently than Maple. Denote the alternate function as altLaguerreL(n, a, x). It is defined as follows:
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For general positive integer a, the relationship for Maple's LaguerreL and altLaguerreL is the following.
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Laguerre polynomials satisfy the following recurrence relation:
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For n not equal to a non-negative integer, the analytic extension of the Laguerre polynomial is given by:
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