FLINT3.7.0-dev
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acb_hypgeom_bessel_k_0f1_series

Synopsis#include <flint/acb_hypgeom.h>
void acb_hypgeom_bessel_k_0f1_series(acb_poly_t res, const acb_poly_t nu, const acb_poly_t z, int scaled, slong len, slong prec)source

Sets res to Kν(z)K_\nu(z) (or ezKν(z)e^z K_\nu(z) if scaled is nonzero) as a power series truncated to length len, for power series ν,z∈C[[x]]\nu, z \in \mathbb{C}[[x]], using

Kν(z)=12πsin⁡(πν)[(z2)−ν0F~1(1−ν,z24)−(z2)ν0F~1(1+ν,z24)].K_{\nu}(z) = \frac{1}{2} \frac{\pi}{\sin(\pi \nu)} \left[ \left(\frac{z}{2}\right)^{-\nu} {}_0{\widetilde F}_1\left(1-\nu, \frac{z^2}{4}\right) - \left(\frac{z}{2}\right)^{\nu} {}_0{\widetilde F}_1\left(1+\nu, \frac{z^2}{4}\right) \right].

If the constant term of ν\nu is an integer, one extra derivative is computed and the singularity is removed (the coefficient of xx in ν\nu must then be nonzero). The output is indeterminate if the constant term of ν\nu is inexact and contains an integer.

Cite this page

The FLINT team, “acb_hypgeom_bessel_k_0f1_series (acb_hypgeom.h),” FLINT: Fast Library for Number Theory, documentation, version 3.7.0-dev, 2026. https://flintlib.org/doc/acb_hypgeom/acb_hypgeom_bessel_k_0f1_series.html (generated 2026-11-02).