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Reduction System

MeijerG.jl includes a reduction system through the main public function meijerg(...). This function attempts to recognize special cases and reduce them to elementary or well-known special functions, then falls back to the pure residue-expansion evaluator meijerg_slater(...) if no match is found.

Overview

The Meijer G-function is extremely general and encodes many classical special functions as special cases. When these cases are recognized, direct evaluation via existing functions (exponential, trigonometric, Bessel, etc.) is typically faster and more numerically accurate than generic residue summation.

The reduction system currently handles the following categories:

  1. Order cancellation: Remove redundant parameters
  2. Elementary functions: Exponential, sine, cosine
  3. Bessel-family reductions: Bessel J, Bessel K, and the order-0 I continuation
  4. Error/incomplete-gamma reductions: erf, erfc, lower gamma, upper gamma
  5. Orthogonal polynomials: Laguerre, Hermite, Jacobi
  6. Logarithmic confluent-pole reduction: log1p(z)/z

API: meijerg

result = meijerg(a, b, m, n, z)
result = meijerg(a_left, a_right, b_left, b_right, z)

If a reduction rule matches, the result is returned directly. If a confluent-pole case is detected, meijerg delegates to the private perturbation routine. Otherwise, meijerg delegates to meijerg_slater for pure residue evaluation.

Category 1: Order Cancellation

When parameters appear in opposite active/inactive partitions, the corresponding $\Gamma$ factors cancel identically before evaluation.

  • If $a_k = b_j$ for some $k \le n$ and $j > m$, remove both parameters and decrement $(p, q, n)$ by one.
  • If $a_k = b_j$ for some $k > n$ and $j \le m$, remove both parameters and decrement $(p, q, m)$ by one.

The reduction is applied recursively, so cancellation cascades are handled automatically.

Category 2: Elementary Functions

Exponential: $e^x$

\[G_{0,1}^{1,0}\!\left(z\;\middle|\;\begin{matrix} - \\ 0 \end{matrix}\right) = e^{-z}\]

Mapping: meijerg((), (), (0,), (), -x) → exp(x)

Sine: $\sin(x)$

\[G_{0,2}^{1,0}\!\left(z\;\middle|\;\begin{matrix} - \\ 1/2, 0 \end{matrix}\right) = \frac{\sin(2\sqrt{z})}{\sqrt{\pi}}\]

Mapping: meijerg((), (), (0.5,), (0,), z^2/4) with scaling → sin(z)

Concretely: $\sin(y) = \sqrt{\pi} \cdot G_{0,2}^{1,0}\!\left(\frac{y^2}{4}\;\middle|\;\begin{matrix} - \\ 1/2, 0 \end{matrix}\right)$

Cosine: $\cos(x)$

\[G_{0,2}^{1,0}\!\left(z\;\middle|\;\begin{matrix} - \\ 0, 1/2 \end{matrix}\right) = \frac{\cos(2\sqrt{z})}{\sqrt{\pi}}\]

Mapping: meijerg((), (), (0,), (0.5,), z^2/4) with scaling → cos(z)

\[\cos(y) = \sqrt{\pi} \cdot G_{0,2}^{1,0}\!\left(\frac{y^2}{4}\;\middle|\;\begin{matrix} - \\ 0, 1/2 \end{matrix}\right)\]

Category 3: Bessel Functions

Bessel J: $J_\nu(x)$

\[G_{0,2}^{1,0}\!\left(z\;\middle|\;\begin{matrix} - \\ \nu/2, -\nu/2 \end{matrix}\right) = J_\nu(2\sqrt{z})\]

Mapping: meijerg((), (), (ν/2,), (-ν/2,), z) → besselj(ν, 2*sqrt(z))

This rule is applied only for strictly positive real z on the real path. Use z + 0im to force complex-branch evaluation.

Bessel I (order-0 continuation on negative reals)

For the special order-zero case, the implementation uses

\[G_{0,2}^{1,0}\!\left(z\;\middle|\;\begin{matrix} - \\ 0, 0 \end{matrix}\right) = I_0(2\sqrt{-z}), \quad z < 0 \text{ real}\]

Mapping: meijerg((), (), (0, 0), (), z) for negative real z → besseli(0, 2*sqrt(-z))

Modified Bessel K: $K_\nu(x)$

\[G_{0,2}^{2,0}\!\left(z\;\middle|\;\begin{matrix} - \\ \nu/2, -\nu/2 \end{matrix}\right) = 2K_\nu(2\sqrt{z})\]

The implementation supports the symmetric identity and the shifted-parameter form

\[G_{0,2}^{2,0}\!\left(z\;\middle|\;\begin{matrix} - \\ b_1, b_2 \end{matrix}\right) = z^{-c} 2K_\nu(2\sqrt{z}), \quad c = \frac{b_1+b_2}{2},\; \nu = |b_1-b_2|\]

Mapping: meijerg((), (), (b1, b2), (), z) → z^(-c) * 2*besselk(ν, 2*sqrt(z))

Category 4: Error Functions and Incomplete Gamma

Error Function: $\mathrm{erf}(x)$

\[G_{1,2}^{1,1}\!\left(z\;\middle|\;\begin{matrix}1\\1/2,0\end{matrix}\right)=\sqrt{\pi}\,\mathrm{erf}(\sqrt{z})\]

Mapping: meijerg((1,), (), (0.5,), (0,), z) → sqrt(pi)*erf(sqrt(z))

Complementary Error Function: $\mathrm{erfc}(x)$

\[G_{1,2}^{2,1}\!\left(z\;\middle|\;\begin{matrix}1\\1/2,0\end{matrix}\right)=\sqrt{\pi}\,\mathrm{erfc}(\sqrt{z})\]

Mapping: meijerg((1,), (), (0.5, 0), (), z) → sqrt(pi)*erfc(sqrt(z))

Lower Incomplete Gamma: $\gamma(a,z)$

\[G_{1,2}^{1,1}\!\left(z\;\middle|\;\begin{matrix}1\\a,0\end{matrix}\right)=\gamma(a,z)\]

Mapping: meijerg((1,), (), (a,), (0,), z) → gamma(a) - gamma(a, z)

Upper Incomplete Gamma: $\Gamma(a,z)$

\[G_{1,2}^{2,1}\!\left(z\;\middle|\;\begin{matrix}1\\a,0\end{matrix}\right)=\Gamma(a,z)\]

Mapping: meijerg((1,), (), (a, 0), (), z) → gamma(a, z)

Category 5: Orthogonal Polynomials

Generalized Laguerre: $L_n^{(\alpha)}(z)$

\[G_{1,2}^{1,0}\!\left(z\;\middle|\;\begin{matrix}n+1\\0,-\alpha\end{matrix}\right) = \frac{L_n^{(\alpha)}(z)}{\Gamma(n+\alpha+1)},\quad n\in\mathbb{Z}_{\ge 0}\]

Mapping: meijerg((n+1,), (), (0,), (-α,), z) → laguerrel(n, α, z) / gamma(n+α+1)

Hermite (via Laguerre specializations)

The implementation recognizes the two half-order channels:

  • even channel: b = (0, 1/2)
  • odd channel: b = (0, -1/2)

and maps through Laguerre forms consistent with standard Hermite-Laguerre identities.

Jacobi: $P_n^{(\alpha,\beta)}(x)$

\[P_n^{(\alpha,\beta)}(x) =\Gamma(n+\alpha+1)\Gamma(-n-\alpha-\beta) G_{2,2}^{1,0}\!\left(\frac{x-1}{2}\;\middle|\;\begin{matrix}n+1,-n-\alpha-\beta\\0,-\alpha\end{matrix}\right)\]

With x = 2z + 1 in the implementation, meijerg((n+1, -n-α-β), (0, -α), 1, 0, z) maps to the corresponding jacobip form.

Category 6: Logarithmic Functions (Confluent Poles)

Confluent-pole Meijer G-functions arise when parameters have integer differences, leading to logarithmic singularities in the residue expansion. For recognized cases, the reduction system uses direct special-function formulas instead of the generic perturbation path.

Logarithm of (1+z): $\log(1 + z) / z$

\[G_{2,2}^{1,2}\!\left(z\;\middle|\;\begin{matrix} 1, 1 \\ 1, 0 \end{matrix}\right) = \frac{\log(1+z)}{z}\]

Mapping: meijerg((1, 1), (1, 0), 1, 2, z) → log1p(z) / z

What about the many G → hypergeometric reductions?

The Meijer G-function has many reduction formulas to generalized hypergeometric functions ${}_pF_q$ or linear combinations thereof. However, the residue expansion used in this package already is this hypergeometric reduction.

To see why, consider the lower expansion for a simple case like $G_{0,2}^{1,0}(z \mid -;\, \nu/2, -\nu/2)$ (related to Bessel J). With $m=1$, the residue loop executes once ($k=1$) and produces:

\[\text{result} = A_1 \cdot z^{b_1} \cdot {}_{0}F_1\!\left(\begin{array}{c}-\\ \beta_1\end{array}\;\middle|\; (-1)^{p-m-n}z\right)\]

That single pFq(α, β, argument) call is the hypergeometric representation found in many formula collections, derived analytically from the pole structure.

References

  • DLMF: https://dlmf.nist.gov/16.17 (Meijer G-function, special cases)
  • Slater, L.J. (1966). Generalized Hypergeometric Functions. Cambridge University Press.