Laguerre Functions
The generalized Laguerre polynomials are defined in terms of confluent
hypergeometric functions as \(L^a_n(x) = ((a+1)_n / n!) {}_1F_1(-n,a+1,x)\),
and are sometimes referred to as the associated Laguerre polynomials.
They are related to the plain Laguerre polynomials \(L_n(x)\) by
\(L^0_n(x) = L_n(x)\) and \(L^k_n(x) = (-1)^k (d^k/dx^k) L_{n+k}(x)\).
For more information see Abramowitz & Stegun, Chapter 22.
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gsl_sf_laguerre_1(a, x)
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gsl_sf_laguerre_2(a, x)
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gsl_sf_laguerre_3(a, x)
These routines evaluate the generalized Laguerre polynomials
\(L^a_1(x), L^a_2(x), L^a_3(x)\) using explicit representations.
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gsl_sf_laguerre_n(n, a, x)
This routine evaluates the generalized Laguerre polynomials
\(L^a_n(x)\) for \(a > -1, n >= 0\).