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  "body": "\n## Definition\n\nLet $q \\in \\C$ with $0 \u003c \\abs{q} \u003c 1$, let $r, s \\ge 0$ be integers, and let $a_1, \\dots, a_r, b_1, \\dots, b_s \\in \\C$ with $b_j \\ne q^{-m}$ for all $j$ and all integers $m \\ge 0$.\nWrite $(c_1, \\dots, c_k; q)_n = (c_1; q)_n \\cdots (c_k; q)_n$ with the [[q-Pochhammer symbol|Y1X8ZZQ7VM]].\nThe *basic hypergeometric series* is the [[power series|0MZB06ST2B]] in $z$\n$${}_r\\phi_s \\left( \\begin{matrix} a_1, \\dots, a_r \\\\ b_1, \\dots, b_s \\end{matrix} ; q, z \\right) = \\sum_{n=0}^{\\infty} \\frac{(a_1, \\dots, a_r; q)_n}{(b_1, \\dots, b_s, q; q)_n} \\left( (-1)^n q^{\\binom{n}{2}} \\right)^{1 + s - r} z^n.$$\nThe *bilateral basic hypergeometric series* is\n$${}_r\\psi_s \\left( \\begin{matrix} a_1, \\dots, a_r \\\\ b_1, \\dots, b_s \\end{matrix} ; q, z \\right) = \\sum_{n=-\\infty}^{\\infty} \\frac{(a_1, \\dots, a_r; q)_n}{(b_1, \\dots, b_s; q)_n} \\left( (-1)^n q^{\\binom{n}{2}} \\right)^{s - r} z^n,$$\nwhere $(a; q)_{-n} = 1 / (a q^{-n}; q)_n$ for $n \\ge 1$.\n\n## Remarks\n\nFor $r = s + 1$ the series ${}_r\\phi_s$ converges for $\\abs{z} \u003c 1$; for $r \\le s$ it converges for all $z \\in \\C$.\nIf some $a_i = q^{-N}$ with an integer $N \\ge 0$, then $(a_i; q)_n = 0$ for $n \u003e N$, and the series terminates.\nThe factor $\\left( (-1)^n q^{\\binom{n}{2}} \\right)^{1+s-r}$ makes the limit $a_r \\to \\infty$ of ${}_r\\phi_s(\\dots; q, z/a_r)$ a series ${}_{r-1}\\phi_s$.\n\nExample: ${}_1\\phi_0(a; -; q, z) = \\sum_{n \\ge 0} \\frac{(a; q)_n}{(q; q)_n} z^n$, and here the ratio of the terms of index $n+1$ and $n$ is $\\frac{1 - a q^n}{1 - q^{n+1}}\\, z$, a rational function of $q^n$.\nBy the [[q-binomial theorem|VVT27RB46B]] it equals $\\frac{(az; q)_\\infty}{(z; q)_\\infty}$ for $\\abs{z} \u003c 1$.\n\nNon-example: $\\sum_{n \\ge 0} \\frac{z^n}{n!}$ is a hypergeometric series but not a basic one, since the ratio $\\frac{z}{n+1}$ is not a rational function of $q^n$.\n\nAs $q \\to 1$, $\\frac{(q^a; q)_n}{(1 - q)^n}$ tends to the Pochhammer symbol $(a)_n$, so ${}_{s+1}\\phi_s(q^{a_1}, \\dots; q^{b_1}, \\dots; q, z)$ tends to ${}_{s+1}F_s(a_1, \\dots; b_1, \\dots; z)$.\n",
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