arb_hypgeom_rising_ui_forward, arb_hypgeom_rising_ui_bs, arb_hypgeom_rising_ui_rs, arb_hypgeom_rising_ui_rec, arb_hypgeom_rising_ui, arb_hypgeom_rising

The variants differ in the algorithm; see Rising factorials.

• arb_hypgeom_rising_ui_forward multiplies the factors one by one at a working precision of prec plus $\log_2 n$ bits, incrementing the factor in place.
• arb_hypgeom_rising_ui_bs multiplies the factors in a balanced binary tree, which is efficient when $x$ has few bits compared to prec (see Binary splitting and product trees).
• arb_hypgeom_rising_ui_rs uses rectangular splitting: it groups the factors in blocks of $m$, expands each block $(x+k)(x+k+1)\cdots(x+k+m-1)$ as a polynomial in $x$ with integer coefficients (one or two words each when they fit, see _arb_hypgeom_rising_coeffs_1) and evaluates it as a dot product with the precomputed powers $x, \ldots, x^m$. Most multiplications of two full-precision numbers become multiplications by small integers [Smi2001; Joh2014b]. The tuning parameter $m$ can be given as 0 to choose it automatically (for large $n$, $m = \min(\sqrt n, 8 + 0.2 \max(0, \mathit{prec} - 4096)^{0.4}, 60)$).
• arb_hypgeom_rising_ui_rec chooses between the three by $n$, prec and the bit length of $x$: the forward recurrence for small $n$ (up to 20 below 512 bits, up to $\min(80, 6000/\mathit{prec})$ otherwise), binary splitting for $n \ge 20$ if $x$ has fewer than $\mathit{prec}/8$ bits, and rectangular splitting otherwise. It never calls the gamma function, which is why the gamma function itself uses it for argument reduction.
• arb_hypgeom_rising_ui uses arb_hypgeom_rising_ui_rec if $n < \max(\mathit{prec}, 100)$, and computes $\Gamma(x+n) \cdot (1/\Gamma(x))$ otherwise. arb_hypgeom_rising does the same for an exact nonnegative integer $n$ in that range and uses the gamma function otherwise.

These are the functions behind arb_rising_ui and arb_rising of arb.

@misc{flint-doc-arb-hypgeom-arb-hypgeom-rising-ui-forward,
  author       = {{The FLINT team}},
  title        = {{arb\_hypgeom\_rising\_ui\_forward, arb\_hypgeom\_rising\_ui\_bs, arb\_hypgeom\_rising\_ui\_rs, arb\_hypgeom\_rising\_ui\_rec, arb\_hypgeom\_rising\_ui, arb\_hypgeom\_rising (arb\_hypgeom.h)}},
  howpublished = {FLINT: Fast Library for Number Theory, documentation, version 3.7.0-dev},
  year         = {2026},
  url          = {https://flintlib.org/doc/arb_hypgeom/arb_hypgeom_rising_ui_forward.html},
  note         = {Generated 2026-11-02}
}