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  "body": "\n## Summary\n\nA basic hypergeometric series, or $q$-hypergeometric series, is a power series $\\sum_n c_n z^n$ in which the ratio $c_{n+1}/c_n$ is a rational function of $q^n$.\nIt is written ${}_r\\phi_s(a_1, \\dots, a_r; b_1, \\dots, b_s; q, z)$ and is built from [[q-Pochhammer symbols|Y1X8ZZQ7VM]].\nIt is the [[q-analog|GFSDMJHHVA]] of the [[generalized hypergeometric series|VDBAW9PN88]], to which it tends as $q \\to 1$ after a change of parameters.\nBasic hypergeometric series contain the partition generating functions, the theta products and the $q$-orthogonal polynomials of the Askey scheme.\n\n## Explanations\n\n### L1 human\n\nIn an ordinary hypergeometric series, the ratio of two consecutive coefficients is a rational function of the index $n$.\nIn a basic hypergeometric series, the index $n$ appears only through the power $q^n$ of a fixed number $q$.\nThe simplest example is $\\sum_{n \\ge 0} \\frac{z^n}{(1-q)(1-q^2)\\cdots(1-q^n)}$, which equals $\\prod_{k \\ge 0} \\frac{1}{1 - z q^k}$ for $\\abs{q} \u003c 1$ and $\\abs{z} \u003c 1$.\nWith $z = q$, it counts the partitions of each number, by the [[generating function for partitions|0578PA53VA]].\n",
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