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  "body": "\n## Definition\n\nFix integers $0 \\le m \\le q$ and $0 \\le n \\le p$, complex numbers $a_1, \\dots, a_p$, $b_1, \\dots, b_q$, and positive reals $A_1, \\dots, A_p$, $B_1, \\dots, B_q$.\nLet $H(z)$ be the [[Fox H-function|EAZMKENA4N]] $H^{m,n}_{p,q}$ with the parameter pairs $(a_j, A_j)$ and $(b_j, B_j)$.\nLet $f \\colon (0, \\infty) \\to \\C$ be such that the integral below converges for $y$ in a set $Y \\subseteq (0, \\infty)$.\nThe *H-transform* of $f$ with these parameters is the function\n$$g(y) = \\int_0^\\infty H(xy)\\, f(x)\\, dx, \\qquad y \\in Y.$$\n\n## Remarks\n\nThe [[Mellin transform|N5PVGRJE1T]] of the kernel is\n$$\\mathcal{M} H(s) = \\frac{\\prod_{j=1}^{m} \\Gamma(b_j + B_j s) \\prod_{j=1}^{n} \\Gamma(1 - a_j - A_j s)}{\\prod_{j=m+1}^{q} \\Gamma(1 - b_j - B_j s) \\prod_{j=n+1}^{p} \\Gamma(a_j + A_j s)},$$\nand formally $\\mathcal{M} g(s) = \\mathcal{M} H(s)\\, \\mathcal{M} f(1 - s)$.\n\nExample: with all $A_j = B_j = 1$, the kernel is a Meijer G-function and the H-transform is a [[Meijer G-function transform|C0BCHX106R]].\nIn particular $m = q = 1$, $n = p = 0$, $b_1 = 0$, $B_1 = 1$ gives the kernel $e^{-z}$, whose Mellin transform is $\\Gamma(s)$, and the H-transform is the [[Laplace transform|0V49JEZW06]].\n\nExample with a scale different from $1$: $m = q = 1$, $n = p = 0$, $b_1 = 0$, $B_1 = 1/2$ gives the kernel with Mellin transform $\\Gamma(s/2)$, which is $2 e^{-z^2}$.\nThe check is $\\int_0^\\infty 2 e^{-z^2} z^{s-1}\\, dz = \\Gamma(s/2)$, by the substitution $u = z^2$.\n\nNon-example: if some $A_j$ or $B_j$ is $0$ or negative, the parameters are outside the definition of the H-function.\n",
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