FLINT3.7.0-dev
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acb_hypgeom_laguerre_l

Synopsis#include <flint/acb_hypgeom.h>
void acb_hypgeom_laguerre_l(acb_t res, const acb_t n, const acb_t m, const acb_t z, slong prec)source

Sets res to the Laguerre polynomial (or Laguerre function)

Lnm(z)=(m+1)nΓ(n+1) 1F1(−n,m+1,z).L_n^{m}(z) = \frac{(m+1)_n}{\Gamma(n+1)}\, {}_1F_1\left(-n,m+1,z\right).

For nonnegative integer nn this is a polynomial in mm and zz, also where the hypergeometric series is undefined; it is then evaluated by direct methods.

There are at least two incompatible extensions to negative integer nn: for m=0m = 0 one can define L−n0(z)=ezLn−10(−z)L_{-n}^0(z) = e^z L_{n-1}^0(-z), or one can use the recurrence Ln−1m(z)+Lnm−1(z)=Lnm(z)L_{n-1}^m(z) + L_n^{m-1}(z) = L_n^m(z). This case is left undefined: the result is indeterminate.

Cite this page

The FLINT team, “acb_hypgeom_laguerre_l (acb_hypgeom.h),” FLINT: Fast Library for Number Theory, documentation, version 3.7.0-dev, 2026. https://flintlib.org/doc/acb_hypgeom/acb_hypgeom_laguerre_l.html (generated 2026-11-02).