Hypergeometric0F1Regularized

The regularized confluent hypergeometric limit function ${}_0F_1(b; z) / \Gamma(b)$ — entire in both $b$ and $z$, unlike Hypergeometric0F1 itself. Provided by @enumeratio/analytic.

✓ checked against that system · ! checked, with disagreements (hover a chip)FungrimHypergeometric0F1Regularized↗Fungrim entry00dfd1↗325a0e↗81eec6↗98703d↗b049dc↗ecd36f↗Wolfram LanguageHypergeometric0F1Regularized!↗

Hypergeometric0F1Regularized(b, z)${}_0F_1(b; z) / \Gamma(b)$.

SymPyhyper([], [b], z) / gamma(b)mpmathenumeratio_hyperreg([], [b], z)

Typed signatures

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

Domain: Special functions

Details

• Computed by its own series $\sum_{k\ge0} z^k / (\Gamma(b+k)\,k!)$ rather than dividing Hypergeometric0F1 by Gamma(b): at $b$ a nonpositive integer, Gamma(b) is itself a pole, and $1/\Gamma$ is taken directly (zero there, by the standard convention) so the sum stays finite exactly where the naive division would not.
• Entire in $z$ too ($p \le q$ for this series), so — unlike Hypergeometric2F1Regularized — never declines on $z$.
• N(x, d) past a double's digits runs a BigDecimal series for real arguments, to $d$ digits; a complex argument stays unevaluated there rather than print a double's digits as if they were $d$.

Plot

Examples

13 more kept as data, checked against the oracles

Basic #

matches Hypergeometric0F1(2, 0.5) / Gamma(2) = Hypergeometric0F1(2, 0.5)

live ↗

In$\mathrm{Hypergeometric0F1Regularized}(2, 0.5)$Out$1.271\,723\,456\,312\,136\,5$

$b = -1$ is a pole of $\Gamma$, but the regularized form is finite there

live ↗

In$\mathrm{Hypergeometric0F1Regularized}(-1, 0.5)$Out$0.147\,179\,736\,722\,106\,7$

At the pole $b = -1$ under N

live ↗

In$N(\mathrm{Hypergeometric0F1Regularized}(-1, 2))$Out$3.715\,035\,560\,458\,436\,4$

live ↗

In$\operatorname{Series}(\mathrm{Hypergeometric0F1Regularized}(2, x), x, 0, 6)$Out$\mathrm{Hypergeometric0F1Regularized}(2, 0)+x\mathrm{Hypergeometric0F1Regularized}^{(0, 1)}(2, 0)+\frac{1}{2}\mathrm{Hypergeometric0F1Regularized}^{(0, 2)}(2, 0)x^2+\frac{1}{6}\mathrm{Hypergeometric0F1Regularized}^{(0, 3)}(2, 0)x^3+\frac{1}{24}\mathrm{Hypergeometric0F1Regularized}^{(0, 4)}(2, 0)x^4+\frac{1}{120}\mathrm{Hypergeometric0F1Regularized}^{(0, 5)}(2, 0)x^5+\frac{1}{720}\mathrm{Hypergeometric0F1Regularized}^{(0, 6)}(2, 0)x^6+O\left(x^7\right)$

live ↗

In$\mathrm{FullSimplify}(\operatorname{Normal}(\operatorname{Series}(\mathrm{Hypergeometric0F1Regularized}(2, x), x, ∞, 3)))$Out$\operatorname{Series}(\mathrm{Hypergeometric0F1Regularized}(2, x), x, ∞, 3)$

Scope #

live ↗

In$N(\mathrm{Hypergeometric0F1Regularized}(0, -48), 50)$Out$-0.753\,564\,079\,781\,443\,083\,803\,278\,642\,115\,376\,608\,410\,090\,278\,370\,78$

live ↗

In$N(\mathrm{Hypergeometric0F1Regularized}(4, 0), 25)$Out$0.166\,666\,666\,666\,666\,666\,666\,666\,7$

live ↗

In$\mathrm{MatrixFunction}(\operatorname{\_1}↦\mathrm{Hypergeometric0F1Regularized}(\frac{1}{2}, \operatorname{\_1}), \bigl\lbrack\bigl\lbrack\frac{1}{2}, 0\bigr\rbrack, \bigl\lbrack0, \frac{1}{2}\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack\operatorname{I}_{-\frac{1}{2}}(\sqrt{2})\sqrt[4]{\frac{1}{2}}, 0\bigr\rbrack, \bigl\lbrack0, \operatorname{I}_{-\frac{1}{2}}(\sqrt{2})\sqrt[4]{\frac{1}{2}}\bigr\rbrack\bigr\rbrack$

Properties #

live ↗

In$N(\mathrm{Hypergeometric0F1Regularized}(1/2, 1/2)^2-\frac{1}{2}\mathrm{Hypergeometric0F1Regularized}(3/2, 1/2)^2)$Out$0.318\,309\,886\,183\,792\,753\,65$

Known value · Wolfram MathematicalFunctionData, Hypergeometric0F1Regularized, FunctionalEquations

live ↗

In$N(\mathrm{Hypergeometric0F1Regularized}(1/2, 2)^2-2\mathrm{Hypergeometric0F1Regularized}(3/2, 2)^2)$Out$0.318\,309\,886\,183\,808\,788\,5$

Known value · Wolfram MathematicalFunctionData, Hypergeometric0F1Regularized, FunctionalEquations

Possible issues #

Exact arguments stay unevaluated; N gives the number

live ↗

In$\mathrm{Hypergeometric0F1Regularized}(-1, 2)$Out$\mathrm{Hypergeometric0F1Regularized}(-1, 2)$

About

Examples	24
Checked against	Wolfram Language 20 agree1 differ1 unevaluatedSymPy 9 agree7 differ7 unevaluatedmpmath 20 agree1 unevaluated
Crosswalk	FungrimFungrim entry ×6Wolfram LanguageSymPympmath

Implementation

primitive · numericirreducibly numeric — it evaluates rather than rewrites

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

mappedexternalwolfram / mpmathpackages/oracle/src/mappings.ts

Hypergeometric0F1Regularized[b,z]; mpmath's own regularized series (rgamma per term).

mappedmpmathpackages/oracle/src/mappings.ts

Regularized by the limit at a pole of Γ(b), through hypercomb.

mappedsympypackages/oracle/src/mappings.ts

SymPy has no regularized hypergeometric, so this divides by Γ(b); at a pole of Γ(b) (b a non-positive integer) the quotient is not the limit that ours is.

See also: Hypergeometric0F1, BesselJ