Hypergeometric2F1Regularized

The regularized Gauss hypergeometric function ${}_2F_1(a,b,c;z) / \Gamma(c)$ (fungrim:fe6e74) — entire in $c$; only converges for $|z| < 1$. Provided by @enumeratio/analytic.

✓ checked against that system · ! checked, with disagreements (hover a chip)FungrimHypergeometric2F1Regularized↗Fungrim entryWolfram LanguageHypergeometric2F1Regularized!↗

Hypergeometric2F1Regularized(a, b, c, z)${}_2F_1(a,b,c;z) / \Gamma(c)$.

Typed signatures

Type	Package	Applies
(number, number, number, number) -> number	analytic	always

Domain: Special functions

Details

The regularized Gauss hypergeometric function $\mathbf{F}(a,b;c;z) = {}_2F_1(a,b;c;z)/\Gamma(c)$ (DLMF 15.2.2, fungrim:fe6e74) divides Hypergeometric2F1 by $\Gamma(c)$, but — unlike doing that division after the fact — is entire in $c$: it stays finite exactly where $\Gamma(c)$ has a pole, at $c$ a nonpositive integer. At such a $c = -n$ it reduces to an ordinary (unregularized) hypergeometric function at shifted parameters, $\mathbf{F}(a,b;-n;z) = \dfrac{(a)_{n+1}(b)_{n+1}}{(n+1)!}\,z^{n+1}\,{}_2F_1(a{+}n{+}1,b{+}n{+}1;n{+}2;z)$ (DLMF 15.2.3_5).

• Computed by the $1/\Gamma$-per-term series ($p = q + 1$ here), so it stays finite at $c$ a nonpositive integer rather than dividing Hypergeometric2F1 by Gamma(c)'s pole there.
• The series only converges for $|z| < 1$; outside the unit disc this stays symbolic rather than answering with a guessed analytic continuation. The exception is N(…, d) past a double's digits at real $z < 1$, which maps $z < -\tfrac12$ into the disc by Pfaff's transformation, $\mathbf{F}(a,b;c;z) = (1-z)^{-a}\,\mathbf{F}(a,c-b;c;\tfrac{z}{z-1})$ (DLMF 15.8.1), and declines at $z \ge 1$ and where the series cannot settle ($z$ near $-\infty$). Several of Fungrim's own identities for this head (e.g. fungrim:90ac58) rewrite to a different argument first — that rewrite belongs in the identity layer, not here.

Examples

5 more kept as data, checked against the oracles

Basic #

matches Hypergeometric2F1(1, 1, 2, 0.5) / Gamma(2)

live ↗

In$\mathrm{Hypergeometric2F1Regularized}(1, 1, 2, 0.5)$Out$1.386\,294\,361\,119\,889\,5$

live ↗

In$\mathrm{Simplify}(\operatorname{Normal}(\operatorname{Series}(\mathrm{Hypergeometric2F1Regularized}(3, 1/2, 1/5, x), x, ∞, 3)))$Out$\operatorname{Series}(\mathrm{Hypergeometric2F1Regularized}(3, \frac{1}{2}, \frac{1}{5}, x), x, ∞, 3)$

live ↗

In$\operatorname{Normal}(\operatorname{Series}(\mathrm{Hypergeometric2F1Regularized}(\frac{1}{2}, 1, 2, x), x, 1, 2))$Out$\frac{1}{2}(\mathrm{Hypergeometric2F1Regularized}^{(0, 0, 0, 2)}(\frac{1}{2}, 1, 2, 1)(x-1)^2)+(x-1)\mathrm{Hypergeometric2F1Regularized}^{(0, 0, 0, 1)}(\frac{1}{2}, 1, 2, 1)+\mathrm{Hypergeometric2F1Regularized}(\frac{1}{2}, 1, 2, 1)$

Scope #

live ↗

In$N(\mathrm{Hypergeometric2F1Regularized}(\frac{1}{3}, 1, 3, -7), 50)$Out$0.355\,102\,040\,816\,326\,530\,612\,244\,897\,959\,183\,673\,469\,387\,755\,102\,04$

live ↗

In$N(\mathrm{Hypergeometric2F1Regularized}(\frac{1}{3}, 1, 0, -7), 25)$Out$-0.145\,8\overline{3}$

live ↗

In$\mathrm{MatrixFunction}(()↦\mathrm{Hypergeometric2F1Regularized}(\frac{1}{2}, 1, 1, \operatorname{\_1}), \bigl\lbrack\bigl\lbrack\frac{1}{2}, 0\bigr\rbrack, \bigl\lbrack0, \frac{1}{2}\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack\sqrt{2}, 0\bigr\rbrack, \bigl\lbrack0, \sqrt{2}\bigr\rbrack\bigr\rbrack$

live ↗

In$\mathrm{Hypergeometric2F1Regularized}(1, 1, 1, 0)$Out$1$

live ↗

In$\mathrm{Hypergeometric2F1Regularized}(\frac{1}{2}, 1, \frac{1}{2}, x)$Out$\frac{1}{(1-x)\sqrt{π}}$

Properties #

stays finite at a nonpositive-integer lower parameter, since $1/\Gamma$ vanishes there

live ↗

In$N(\mathrm{Hypergeometric2F1Regularized}(1, 1, -1, 0.3))$Out$0.524\,781\,341\,107\,871\,2$

Possible issues #

$|z| \ge 1$ stays symbolic — no continuation past the unit disc

live ↗

In$\mathrm{Hypergeometric2F1Regularized}(1, 1, 2, 1.5)$Out$\mathrm{Hypergeometric2F1Regularized}(1, 1, 2, 1.5)$

About

Examples	15
Checked against	Wolfram Language 12 agree1 differ2 unevaluated
Crosswalk	FungrimFungrim entry ×9Wolfram Language

Implementation

primitive · numericirreducibly numeric — it evaluates rather than rewrites

nativeenginetypescriptpackages/symbols/analysis/analytic/src/hypergeometric.ts

mappedexternalwolfram / mpmathpackages/oracle/src/mappings.ts

Hypergeometric2F1Regularized[a,b,c,z]; mpmath's own regularized series (rgamma per term).

See also: Hypergeometric3F2Regularized, Hypergeometric1F1Regularized