Hypergeometric0F1

The confluent hypergeometric limit function ${}_0F_1(b; z) = \sum_{k\ge0} z^k / ((b)_k\,k!)$ — entire in $z$, related to the Bessel functions. Provided by @enumeratio/analytic.

✓ checked against that system · ! checked, with disagreements (hover a chip)FungrimHypergeometric0F1✓↗Fungrim entryWolfram LanguageHypergeometric0F1!↗

Hypergeometric0F1(b, z)${}_0F_1(b; z)$, by its defining series.

SageMathhypergeometric([], [b], z)SymPyN(hyper([], [b], z))mpmathhyp0f1(b, z)

Typed signatures

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

Domain: Special functions

Details

The confluent hypergeometric limit function ${}_0F_1(b; z) = \sum_{k\ge0} z^k / ((b)_k\,k!)$ is the $p=0$, $q=1$ case of HypergeometricPFQ (DLMF 16.2.1) and, having no upper parameter, is entire in $z$ for every $b$. It is Bessel's equation in series form: $J_\nu(z) = \dfrac{(z/2)^\nu}{\Gamma(\nu+1)}\,{}_0F_1(\nu+1; -z^2/4)$ (DLMF 10.16.9), and likewise for $I_\nu$ with $z^2/4$ in place of $-z^2/4$.

• compute-engine 0.128 does not declare this head at all — no Hypergeometric1F1-style native to extend — so it is supplied here directly, by the term-ratio recurrence (Fungrim's own Hypergeometric0F1 identities relate it to AiryAi, Sin and Sinc).
• Poles at $b$ a nonpositive integer $0, -1, -2, \dots$ stay symbolic; see Hypergeometric0F1Regularized for the entire version.

Plot

Examples

6 more kept as data, checked against the oracles

Basic #

a generic point, checked against mpmath's hyp0f1

live ↗

In$\mathrm{Hypergeometric0F1}(2, 0.5)$Out$1.271\,723\,456\,312\,137$

live ↗

In$\operatorname{Series}(\mathrm{Hypergeometric0F1}(a, x), x, 0, 4)$Out$1+\frac{x}{a}+\frac{x^2}{2a×(a+1)}+\frac{x^3}{6a×(a+1)(a+2)}+\frac{x^4}{24a×(a+1)(a+2)(a+3)}+O\left(x^5\right)$

live ↗

In$\mathrm{FullSimplify}(\operatorname{Normal}(\operatorname{Series}(\mathrm{Hypergeometric0F1}(a, x), x, ∞, 1)))$Out$\operatorname{Series}(\mathrm{Hypergeometric0F1}(a, x), x, ∞, 1)$

Scope #

live ↗

In$N(\mathrm{Hypergeometric0F1}(1, -2), 50)$Out$-0.196\,548\,095\,270\,468\,200\,040\,793\,372\,087\,932\,231\,325\,889\,787\,310\,89$

live ↗

In$\mathrm{MatrixFunction}(\operatorname{\_1}↦\mathrm{Hypergeometric0F1}(\frac{1}{2}, \operatorname{\_1}), \bigl\lbrack\bigl\lbrack1, 0\bigr\rbrack, \bigl\lbrack0, 1\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack\mathrm{Hypergeometric0F1}(\frac{1}{2}, 1), 0\bigr\rbrack, \bigl\lbrack0, \mathrm{Hypergeometric0F1}(\frac{1}{2}, 1)\bigr\rbrack\bigr\rbrack$

live ↗

In$\operatorname{Series}(\mathrm{Hypergeometric0F1}(a, x), x, 0, 3)$Out$1+\frac{x}{a}+\frac{x^2}{2a×(a+1)}+\frac{x^3}{6a×(a+1)(a+2)}+O\left(x^4\right)$

Properties #

live ↗

In$N(-\frac{1}{4}\mathrm{Hypergeometric0F1}(\frac{1}{2}, \frac{1}{2})+\frac{1}{4}\mathrm{Hypergeometric0F1}(-\frac{1}{2}, \frac{1}{2})+\frac{1}{2}\mathrm{Hypergeometric0F1}(\frac{3}{2}, \frac{1}{2}))$Out$7.5\cdot10^{-17}$

Known value · Wolfram MathematicalFunctionData, Hypergeometric0F1, FunctionalEquations

live ↗

In$N(\frac{1}{2}\mathrm{Hypergeometric0F1}(3/2, 1/2)^2-\frac{1}{4}\mathrm{Hypergeometric0F1}(1/2, 1/2)^2)$Out$-0.249\,999\,999\,999\,999\,574\,855$

Known value · Wolfram MathematicalFunctionData, Hypergeometric0F1, FunctionalEquations

live ↗

In$N(2\mathrm{Hypergeometric0F1}(\frac{3}{2}, 2)-\frac{\mathrm{Hypergeometric0F1}(\frac{1}{2}, 2)}{4}+\frac{\mathrm{Hypergeometric0F1}(-\frac{1}{2}, 2)}{4})$Out$-2.5\cdot10^{-16}$

Known value · Wolfram MathematicalFunctionData, Hypergeometric0F1, FunctionalEquations

live ↗

In$N(2\mathrm{Hypergeometric0F1}(3/2, 2)^2-\frac{1}{4}\mathrm{Hypergeometric0F1}(1/2, 2)^2)$Out$-0.249\,999\,999\,999\,989\,795\,465$

Known value · Wolfram MathematicalFunctionData, Hypergeometric0F1, FunctionalEquations

Possible issues #

$b = 0$ is a pole — stays symbolic

live ↗

In$\mathrm{Hypergeometric0F1}(0, 0.5)$Out$\mathrm{Hypergeometric0F1}(0, 0.5)$

About

Examples	17
Checked against	Wolfram Language 14 agree2 differ1 unevaluatedSageMath 14 agree1 unevaluated1 errorSymPy 14 agree1 differ1 unevaluated1 errormpmath 12 agree1 error
Crosswalk	FungrimFungrim entry ×8Wolfram LanguageSageMathSymPympmath

Implementation

primitive · numericirreducibly numeric — it evaluates rather than rewrites

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

mappedexternalwolfram / mpmathpackages/oracle/src/mappings.ts

Hypergeometric0F1[b,z]; mpmath.hyp0f1(b,z).

mappedsympypackages/oracle/src/mappings.ts

SymPy's hyper stays held at numeric arguments until N, so the template evaluates it.

mappedmpmathpackages/oracle/src/mappings.ts

mappedsagepackages/oracle/src/mappings.ts

See also: Hypergeometric0F1Regularized, BesselJ