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)FungrimHypergeometric0F1RegularizedFungrim entry00dfd1325a0e81eec698703db049dcecd36fWolfram LanguageHypergeometric0F1Regularized!

Hypergeometric0F1Regularized(b, z)

{}_0F_1(b; z) / \Gamma(b)

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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 $d$ .

Examples

10 more kept as data, checked against the oracles

Basic #

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

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

In

\mathrm{Hypergeometric0F1Regularized}(-1, 0.5)

Out

0.147\,179\,736\,722\,106\,7

At the pole $b = -1$ under N

In

N(\mathrm{Hypergeometric0F1Regularized}(-1, 2))

Out

3.715\,035\,560\,458\,436\,4

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)

In

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

Out

\operatorname{Series}(\mathrm{Hypergeometric0F1Regularized}(2, x), x, ∞, 3)

Scope #

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

In

N(\mathrm{Hypergeometric0F1Regularized}(4, 0), 25)

Out

0.1\overline{6}

In

\mathrm{MatrixFunction}(()↦\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 #

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

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

In

\mathrm{Hypergeometric0F1Regularized}(-1, 2)

Out

\mathrm{Hypergeometric0F1Regularized}(-1, 2)

About

Examples	21
Checked against	Wolfram Language 19 agree1 differ1 unevaluated
Crosswalk	Fungrim Fungrim entry ×6 Wolfram Language

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).

See also: Hypergeometric0F1, BesselJ