MeijerG

The Meijer G-function $G^{m,n}_{p,q}$, evaluated numerically via its reduction to a finite sum of ordinary ${}_pF_q$ series (DLMF 16.17.2), run in BigDecimal (meijer-g-big.ts) so the reduction's own cancellation lands a few ulps of the true value rather than ~1e-13.

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

MeijerG({{a1, .., an}, {a(n+1), .., ap}}, {{b1, .., bm}, {b(m+1), .., bq}}, z)$G^{m,n}_{p,q}\left(z \,\middle|\, \begin{matrix}a_1,\ldots,a_p\\b_1,\ldots,b_q\end{matrix}\right)$, numerically.

Typed signatures

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

Domain: Special functions

Details

• Numeric only (like this package's Hypergeometric0F1 and friends — always declines for a symbolic z or parameter, never returns a symbolic closed form). Requires $m \ge 1$ and $p \le q$, so every term's ${}_pF_q$ has upper-count $\le$ lower-count $+ 1$, the shape pfqSeries already handles (entire, or convergent only inside the unit disc).
• Declined: $p > q$; $m = 0$; two of the $b_h$ ($h \le m$) congruent mod 1 (a non-simple pole — the log-case formula this does not implement); a Γ-argument in the prefactor landing on a non-positive integer (a further degeneracy); non-numeric operands.
• Verified against wolframscript's own MeijerG at several parameter sets, including the plain-exponential case ($m{=}1,n{=}0,p{=}0,q{=}1$) and cases with both $a$- and $b$-parameters.

Plot

argument p1 of type list has no control

Examples

31 more kept as data, checked against the oracles

Basic #

$G^{1,0}_{0,1}(z \mid {}; 0) = e^{-z}$

live ↗

In$\mathrm{MeijerG}(\bigl\lbrack\bigl\lbrack\bigr\rbrack, \bigl\lbrack\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack0\bigr\rbrack, \bigl\lbrack\bigr\rbrack\bigr\rbrack, 0.6)$Out$0.548\,811\,636\,094\,026\,4$

Known value · mpmath 1.3.0 meijerg

live ↗

In$N(\mathrm{MeijerG}(\bigl\lbrack\bigl\lbrack\bigr\rbrack, \bigl\lbrack2\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack1/2, 3/2\bigr\rbrack, \bigl\lbrack\bigr\rbrack\bigr\rbrack, 3), 50)$Out$\mathrm{MeijerG}(\bigl\lbrack\bigl\lbrack\bigr\rbrack, \bigl\lbrack2\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack0.5, 1.5\bigr\rbrack, \bigl\lbrack\bigr\rbrack\bigr\rbrack, 3)$

live ↗

In$\operatorname{Series}(\mathrm{MeijerG}(\bigl\lbrack\bigl\lbrack a\bigr\rbrack, \bigl\lbrack\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack d\bigr\rbrack, \bigl\lbrack\bigr\rbrack\bigr\rbrack, x), x, 0, 2)$Out$\mathrm{MeijerG}(\bigl\lbrack\bigl\lbrack a\bigr\rbrack, \bigl\lbrack\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack d\bigr\rbrack, \bigl\lbrack\bigr\rbrack\bigr\rbrack, 0)+x\mathrm{MeijerG}^{(0, 0, 1)}(\bigl\lbrack\bigl\lbrack a\bigr\rbrack, \bigl\lbrack\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack d\bigr\rbrack, \bigl\lbrack\bigr\rbrack\bigr\rbrack, 0)+\frac{1}{2}\mathrm{MeijerG}^{(0, 0, 2)}(\bigl\lbrack\bigl\lbrack a\bigr\rbrack, \bigl\lbrack\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack d\bigr\rbrack, \bigl\lbrack\bigr\rbrack\bigr\rbrack, 0)x^2+O\left(x^3\right)$

Scope #

a case with both an $a$- and a $b$-parameter

live ↗

In$\mathrm{MeijerG}(\bigl\lbrack\bigl\lbrack0.7\bigr\rbrack, \bigl\lbrack\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack0.3\bigr\rbrack, \bigl\lbrack\bigr\rbrack\bigr\rbrack, 0.4)$Out$0.924\,468\,128\,124\,581\,7$

Known value · mpmath 1.3.0 meijerg

$m = 2$, two poles summed

live ↗

In$\mathrm{MeijerG}(\bigl\lbrack\bigl\lbrack\bigr\rbrack, \bigl\lbrack\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack0, \frac{1}{2}\bigr\rbrack, \bigl\lbrack\bigr\rbrack\bigr\rbrack, 1.3)$Out$0.181\,230\,440\,188\,233\,34$

Known value · mpmath 1.3.0 meijerg

A Meijer G value at 3 to 50 digits with the imaginary noise removed; the kernel is a double, so past a double's digits the call stays unevaluated

live ↗

In$\mathrm{chop}(N(\mathrm{MeijerG}(\bigl\lbrack\bigl\lbrack1\bigr\rbrack, \bigl\lbrack2\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack2/3, 1/5\bigr\rbrack, \bigl\lbrack4\bigr\rbrack\bigr\rbrack, 3), 50))$Out$\mathrm{chop}(\mathrm{MeijerG}(\bigl\lbrack\bigl\lbrack1\bigr\rbrack, \bigl\lbrack2\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack2/3, 1/5\bigr\rbrack, \bigl\lbrack4\bigr\rbrack\bigr\rbrack, 3))$

About

Examples	37
Checked against	Wolfram Language 34 agree1 differ1 unevaluatedSymPy 33 agree1 differ1 unevaluated1 errormpmath 33 agree1 differ1 unevaluated
Crosswalk	Wolfram LanguageSymPympmath

Implementation

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

nativeenginetypescriptpackages/symbols/analysis/analytic/src/meijer-g.ts

mappedsympypackages/oracle/src/mappings.ts

Wolfram's {{a1..an}, {an+1..ap}}, {{b1..bm}, {bm+1..bq}} nesting is SymPy's own.

mappedmpmathpackages/oracle/src/mappings.ts

See also: MeijerGReduce, HypergeometricPFQ