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  "body": "\n## Summary\n\nThe generalized hypergeometric function ${}_pF_q$ is a power series whose coefficient ratio is a rational function of $n$.\nThat rational function has $p$ numerator parameters and $q$ denominator parameters.\nThe [[hypergeometric function|463X5V31D9]] ${}_2F_1$ is the case $p = 2$, $q = 1$, and the exponential, the binomial series and the Bessel functions come from smaller $p$ and $q$.\nMany classical summation formulas, such as those of Pfaff–Saalschütz, Dixon and Dougall, are evaluations of ${}_3F_2$ and higher series at $z = 1$.\n\n## Explanations\n\n### L1 human\n\nA series $\\sum t_n$ is called hypergeometric when the ratio $t_{n+1}/t_n$ of consecutive terms is a rational function of $n$.\nAfter factoring numerator and denominator into linear factors, such a series can be written in a standard form ${}_pF_q$ with $p$ upper and $q$ lower parameters.\nFor example, $e^z = \\sum z^n / n!$ has ratio $\\frac{z}{n+1}$, so it is ${}_0F_0(;;z)$.\nThe numbers $p$ and $q$ control growth: the series is entire when $p \\le q$, converges in the unit disc when $p = q + 1$, and diverges when $p \u003e q + 1$, except when it is a polynomial.\n",
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