HypergeometricPFQ

The generalized hypergeometric function ${}_pF_q(a; b; z)$, built on the same series machinery (pfqSeries) as this package's 0F1/1F1Regularized/2F1Regularized/3F2Regularized, generalized to arbitrary $p$ and $q$.

✓ checked against that system · ! checked, with disagreements (hover a chip)Wolfram LanguageHypergeometricPFQ✓↗

HypergeometricPFQ(a, b, z)${}_pF_q(a; b; z)$, for parameter lists a (length $p$) and b (length $q$).

Typed signatures

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

Domain: Special functions

Details

The generalized hypergeometric function ${}_pF_q(a_1,\dots,a_p; b_1,\dots,b_q; z) = \sum_{k\ge0} \dfrac{(a_1)_k \cdots (a_p)_k}{(b_1)_k \cdots (b_q)_k}\dfrac{z^k}{k!}$ (DLMF 16.2.1) is the family this package's other hypergeometric heads specialize: Hypergeometric0F1 is $p=0,q=1$; Hypergeometric2F1Regularized is the $p=2,q=1$ regularized case. Convergence follows from the parameter counts alone (DLMF §16.2(ii)–(iv)): entire in $z$ for $p \le q$, radius 1 for $p=q+1$ (unless an upper parameter terminates the series into a polynomial), and generically divergent for $p > q+1$.

• Three closed forms stay exact ahead of the numeric series, symbolic $z$ (and parameters) included: ${}_0F_0(;;z) = e^z$; ${}_1F_0(a;;z) = (1-z)^{-a}$; and $\mathrm{HypergeometricPFQ}(\ldots; 0) = 1$ for any parameter lists (the series' own leading term).
• Otherwise: $p \le q$ converges for any $z$; $p = q+1$ only inside the unit disc, matching this package's Regularized forms — declined outside it rather than attempting an analytic continuation.

Examples

5 more kept as data, checked against the oracles

Basic #

${}_2F_1(1, 1; 2; z) = -\ln(1-z)/z$, here $2\ln 2$

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In$\mathrm{HypergeometricPFQ}(\bigl\lbrack1, 1\bigr\rbrack, \bigl\lbrack2\bigr\rbrack, 0.5)$Out$1.386\,294\,361\,119\,890\,6$

Known value · Wolfram 15.0.0 HypergeometricPFQ

${}_1F_1(1; 2; z) = (e^z - 1)/z$

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In$\mathrm{HypergeometricPFQ}(\bigl\lbrack1\bigr\rbrack, \bigl\lbrack2\bigr\rbrack, 0.5)$Out$1.297\,442\,541\,400\,256\,4$

Scope #

${}_3F_2$

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In$\mathrm{HypergeometricPFQ}(\bigl\lbrack1, 2, 3\bigr\rbrack, \bigl\lbrack4, 5\bigr\rbrack, 0.5)$Out$1.189\,874\,754\,256\,423$

Known value · Wolfram 15.0.0 HypergeometricPFQ

Properties #

${}_0F_0(;;z) = e^z$

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In$\mathrm{HypergeometricPFQ}(\bigl\lbrack\bigr\rbrack, \bigl\lbrack\bigr\rbrack, z)$Out$e^{z}$

${}_1F_0(a;;z) = (1-z)^{-a}$

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In$\mathrm{HypergeometricPFQ}(\bigl\lbrack a\bigr\rbrack, \bigl\lbrack\bigr\rbrack, z)$Out$(1-z)^{-a}$

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In$\mathrm{HypergeometricPFQ}(\bigl\lbrack\frac{1}{2}\bigr\rbrack, \bigl\lbrack\bigr\rbrack, 2)$Out$-i$

Known value · Wolfram MathematicalFunctionData, HypergeometricPFQ:1F0, FunctionalEquations

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In$\mathrm{HypergeometricPFQ}(\bigl\lbrack\frac{1}{2}\bigr\rbrack, \bigl\lbrack\bigr\rbrack, 3)$Out$-\frac{\sqrt{2}}{2}i$

Known value · Wolfram MathematicalFunctionData, HypergeometricPFQ:1F0, FunctionalEquations

live ↗

In$N(\frac{1}{2}(\mathrm{HypergeometricPFQ}(\bigl\lbrack1/2\bigr\rbrack, \bigl\lbrack1/2, 1/2\bigr\rbrack, \frac{1}{2}))-\frac{1}{2}(\mathrm{HypergeometricPFQ}(\bigl\lbrack3/2\bigr\rbrack, \bigl\lbrack3/2, 1/2\bigr\rbrack, \frac{1}{2})))$Out$0.0$

Known value · Wolfram MathematicalFunctionData, HypergeometricPFQ:1F2, FunctionalEquations

live ↗

In$N(\frac{1}{2}(\mathrm{HypergeometricPFQ}(\bigl\lbrack1/2\bigr\rbrack, \bigl\lbrack3/2, 1/2\bigr\rbrack, \frac{1}{2}))-\frac{1}{2}(\mathrm{HypergeometricPFQ}(\bigl\lbrack1/2\bigr\rbrack, \bigl\lbrack1/2, 3/2\bigr\rbrack, \frac{1}{2})))$Out$0.0$

Known value · Wolfram MathematicalFunctionData, HypergeometricPFQ:1F2, FunctionalEquations

live ↗

In$N(\frac{1}{2}(\mathrm{HypergeometricPFQ}(\bigl\lbrack1/2\bigr\rbrack, \bigl\lbrack1/2, 1/2\bigr\rbrack, \frac{1}{2}))-\frac{1}{2}(\mathrm{HypergeometricPFQ}(\bigl\lbrack3/2\bigr\rbrack, \bigl\lbrack3/2, 1/2\bigr\rbrack, \frac{1}{2})))$Out$0.0$

Known value · Wolfram MathematicalFunctionData, HypergeometricPFQ:1F2, FunctionalEquations

live ↗

In$N(\frac{1}{2}(\mathrm{HypergeometricPFQ}(\bigl\lbrack1/2\bigr\rbrack, \bigl\lbrack3/2, 1/2\bigr\rbrack, \frac{1}{2}))-\frac{1}{2}(\mathrm{HypergeometricPFQ}(\bigl\lbrack1/2\bigr\rbrack, \bigl\lbrack1/2, 3/2\bigr\rbrack, \frac{1}{2})))$Out$0.0$

Known value · Wolfram MathematicalFunctionData, HypergeometricPFQ:1F2, FunctionalEquations

live ↗

In$N(\frac{1}{4}(\mathrm{HypergeometricPFQ}(\bigl\lbrack1/2\bigr\rbrack, \bigl\lbrack1/2, 1/2\bigr\rbrack, 1/2)-\mathrm{HypergeometricPFQ}(\bigl\lbrack3/2\bigr\rbrack, \bigl\lbrack1/2, 1/2\bigr\rbrack, 1/2))+\frac{1}{2}(\mathrm{HypergeometricPFQ}(\bigl\lbrack3/2\bigr\rbrack, \bigl\lbrack3/2, 3/2\bigr\rbrack, \frac{1}{2})))$Out$-2.5\cdot10^{-17}$

Known value · Wolfram MathematicalFunctionData, HypergeometricPFQ:1F2, FunctionalEquations

live ↗

In$N(\frac{1}{8}(3(\mathrm{HypergeometricPFQ}(\bigl\lbrack1/2\bigr\rbrack, \bigl\lbrack1/2, 1/2\bigr\rbrack, 1/2)-\mathrm{HypergeometricPFQ}(\bigl\lbrack1/2\bigr\rbrack, \bigl\lbrack3/2, 1/2\bigr\rbrack, 1/2)))-\frac{1}{4}(\mathrm{HypergeometricPFQ}(\bigl\lbrack3/2\bigr\rbrack, \bigl\lbrack5/2, 3/2\bigr\rbrack, \frac{1}{2})))$Out$-1.0\cdot10^{-16}$

Known value · Wolfram MathematicalFunctionData, HypergeometricPFQ:1F2, FunctionalEquations

live ↗

In$N(\frac{1}{8}(3(\mathrm{HypergeometricPFQ}(\bigl\lbrack1/2\bigr\rbrack, \bigl\lbrack1/2, 1/2\bigr\rbrack, 1/2)-\mathrm{HypergeometricPFQ}(\bigl\lbrack3/2\bigr\rbrack, \bigl\lbrack3/2, 1/2\bigr\rbrack, 1/2))))$Out$0.0$

Known value · Wolfram MathematicalFunctionData, HypergeometricPFQ:1F2, FunctionalEquations

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In$N(\mathrm{HypergeometricPFQ}(\bigl\lbrack-\frac{1}{2}\bigr\rbrack, \bigl\lbrack\frac{1}{2}, \frac{1}{2}\bigr\rbrack, \frac{1}{2})+\mathrm{HypergeometricPFQ}(\bigl\lbrack\frac{1}{2}\bigr\rbrack, \bigl\lbrack\frac{3}{2}, \frac{1}{2}\bigr\rbrack, \frac{1}{2}))$Out$1.310\,463\,412\,812\,374\,734$

Known value · Wolfram MathematicalFunctionData, HypergeometricPFQ:1F2, FunctionalEquations

live ↗

In$N(\frac{1}{2}(\mathrm{HypergeometricPFQ}(\bigl\lbrack1/2\bigr\rbrack, \bigl\lbrack1/2, 1/2\bigr\rbrack, 2))-\frac{1}{2}(\mathrm{HypergeometricPFQ}(\bigl\lbrack3/2\bigr\rbrack, \bigl\lbrack3/2, 1/2\bigr\rbrack, 2)))$Out$0.0$

Known value · Wolfram MathematicalFunctionData, HypergeometricPFQ:1F2, FunctionalEquations

live ↗

In$N(\frac{1}{2}(\mathrm{HypergeometricPFQ}(\bigl\lbrack1/2\bigr\rbrack, \bigl\lbrack3/2, 1/2\bigr\rbrack, 2))-\frac{1}{2}(\mathrm{HypergeometricPFQ}(\bigl\lbrack1/2\bigr\rbrack, \bigl\lbrack1/2, 3/2\bigr\rbrack, 2)))$Out$0.0$

Known value · Wolfram MathematicalFunctionData, HypergeometricPFQ:1F2, FunctionalEquations

About

Examples	22
Checked against	Wolfram Language 22 agree
Crosswalk	Wolfram Language

Implementation

primitive · kernela kernel — an algorithm over mutable state that would not be readable as a tree

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

See also: Hypergeometric0F1, Hypergeometric2F1Regularized, Hypergeometric3F2Regularized