FLINT3.7.0-dev
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acb_hypgeom_chebyshev_t, acb_hypgeom_chebyshev_u

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

Sets res to the Chebyshev polynomial or Chebyshev function of the first or second kind

Tn(z)=2F1(−n,n,12,1−z2),Un(z)=(n+1) 2F1(−n,n+2,32,1−z2).T_n(z) = {}_2F_1\left(-n,n,\frac{1}{2},\frac{1-z}{2}\right), \qquad U_n(z) = (n+1)\, {}_2F_1\left(-n,n+2,\frac{3}{2},\frac{1-z}{2}\right).

For word-size integer nn the functions call acb_chebyshev_t_ui and acb_chebyshev_u_ui. Otherwise the hypergeometric series is used near z=1z = 1, and the trigonometric representations Tn(cos⁡θ)=cos⁡(nθ)T_n(\cos \theta) = \cos(n\theta) and Un(cos⁡θ)=sin⁡((n+1)θ)/sin⁡θU_n(\cos\theta) = \sin((n+1)\theta)/\sin\theta elsewhere.

Cite this page

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