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  "body": "\n## Summary\n\nThe Fox H-function is a generalisation of the [[Meijer G-function|19JNNN5KV7]].\nIn its [[Barnes integral|1PDGBGEHM2]] the gamma factors may have the form $\\Gamma(b + B s)$ with a positive scale $B$, not only $\\Gamma(b + s)$.\nWhen all scales equal $1$, it is a Meijer G-function.\nThe extra freedom covers functions such as the Mittag-Leffler and Fox–Wright functions.\nH-functions appear in fractional calculus and as densities in probability, for instance of products and quotients of independent random variables.\n\n## Explanations\n\n### L1 human\n\nA Meijer G-function is described by a Mellin transform that is a ratio of gamma values at points $b + s$.\nFox allowed points $b + B s$ with any positive stretch $B$.\nThis small change brings in functions whose series have factorials of non-integer multiples of the index, such as $\\sum_k z^k / \\Gamma(\\alpha k + 1)$.\nA basic example is $H^{1, 0}_{0, 1}\\left[ z \\,\\middle|\\, (0, 1) \\right] = e^{-z}$, which comes from the Mellin transform $\\Gamma(s)$ of $e^{-z}$.\n",
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