FLINT3.7.0-dev
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acb_polygamma

Synopsis#include <flint/acb.h>
void acb_polygamma(acb_t res, const acb_t s, const acb_t z, slong prec)source

Sets res to the generalised polygamma function ψ(s,z)\psi(s, z).

For a nonnegative integer ss this is the ss-th derivative of the digamma function: s=0s = 0 calls acb_digamma and integers s≥1s \ge 1 the Hurwitz zeta function (for small ss, reading the coefficients of acb_poly_digamma_series can be faster). Other ss use the generalisation of Espinosa and Moll [EM2004],

ψ(s,z)=ζ′(s+1,z)+(γ+ψ(−s))ζ(s+1,z)Γ(−s).\psi(s, z) = \frac{\zeta'(s+1, z) + (\gamma + \psi(-s)) \zeta(s+1, z)}{\Gamma(-s)}.

References#

  • [EM2004] O. Espinosa and V. Moll, "A generalized polygamma function", Integral Transforms and Special Functions (2004), 101-115.
Cite this page

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