FLINT3.7.0-dev
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acb_hypgeom_2f1_transform, acb_hypgeom_2f1_transform_limit

Synopsis#include <flint/acb_hypgeom.h>
void acb_hypgeom_2f1_transform(acb_t res, const acb_t a, const acb_t b, const acb_t c, const acb_t z, int flags, int which, slong prec)source
void acb_hypgeom_2f1_transform_limit(acb_t res, const acb_t a, const acb_t b, const acb_t c, const acb_t z, int regularized, int which, slong prec)source

Sets res to F(z)F(z) (or F(z)\mathbf{F}(z)) computed with the linear fractional transformation selected by which: 1 for z/(z−1)z/(z-1), 2 for 1/z1/z, 3 for 1/(1−z)1/(1-z), 4 for 1−z1-z and 5 for 1−1/z1-1/z.

Transformation 1 (Pfaff) maps the function to one 2F1{}_2F_1 series; the others express it as a combination of two series with gamma function factors [NIST2012, §15.8], which is singular when b−ab - a (for 2 and 3) or c−a−bc - a - b (for 4 and 5) is an integer. acb_hypgeom_2f1_transform detects this case from the exactness of the parameters or from flags (with the meaning of acb_hypgeom_2f1) and then calls acb_hypgeom_2f1_transform_limit, which assumes that which is not 1, that b−ab - a is an exact integer if which is 2 or 3 and that c−a−bc - a - b is an exact integer if which is 4 or 5, and computes the limit by differentiating with respect to a perturbation of aa (power series of length 2). The result is indeterminate if zz contains 0 or 1.

References#

  • [NIST2012] National Institute of Standards and Technology, Digital Library of Mathematical Functions (2012), https://dlmf.nist.gov/
Cite this page

The FLINT team, “acb_hypgeom_2f1_transform, acb_hypgeom_2f1_transform_limit (acb_hypgeom.h),” FLINT: Fast Library for Number Theory, documentation, version 3.7.0-dev, 2026. https://flintlib.org/doc/acb_hypgeom/acb_hypgeom_2f1_transform.html (generated 2026-11-02).