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  "body": "\n## Statement\n\nLet $a, b, q, z \\in \\C$ with $0 \u003c \\abs{q} \u003c 1$, $a \\ne 0$, $b \\notin \\set{q^{-m} : m \\ge 0}$, and $\\abs{b/a} \u003c \\abs{z} \u003c 1$.\nFor an integer $n \\ge 1$ put $(a; q)_{-n} = 1 / (a q^{-n}; q)_n$, so that $(a; q)_n$ is defined for all $n \\in \\Z$ through the [[q-Pochhammer symbol|Y1X8ZZQ7VM]].\nThen\n$$\\sum_{n=-\\infty}^{\\infty} \\frac{(a; q)_n}{(b; q)_n} z^n = \\frac{(q; q)_\\infty\\, (b/a; q)_\\infty\\, (az; q)_\\infty\\, (q/(az); q)_\\infty}{(b; q)_\\infty\\, (q/a; q)_\\infty\\, (z; q)_\\infty\\, (b/(az); q)_\\infty}.$$\nIn the notation of [[bilateral basic hypergeometric series|EB4N9RQR51]], the left side is ${}_1\\psi_1(a; b; q, z)$.\n\n## Remarks\n\nThe condition $\\abs{z} \u003c 1$ makes the part with $n \\ge 0$ converge, and $\\abs{b/a} \u003c \\abs{z}$ makes the part with $n \u003c 0$ converge.\nProof idea: for $b = q$ the terms with $n \u003c 0$ vanish and the identity is the [[q-binomial theorem|VVT27RB46B]]; then $b = q^{m+1}$ follows by a shift of the summation index.\nBoth sides are analytic in $b$ in a disc, and they agree on the points $q^{m}$, $m \\ge 1$, which accumulate at $0$, so they agree everywhere by the identity theorem.\nExample: with $b = 0$ (a limit) and $z$ replaced by $-z/a$, letting $a \\to \\infty$ turns the identity into the [[Jacobi triple product|JFP2DGMNXT]].\nThe identity is due to Ramanujan, who stated it without proof; Hahn and Jackson gave the first published proofs around 1950.\n\n## Explanations\n\n### L1 human\n\nThis formula sums a two-sided series, running over all integers $n$, whose terms are ratios of $q$-Pochhammer symbols.\nThe answer is a ratio of eight infinite products.\nIt is the two-sided version of the $q$-binomial theorem.\n\n### L2 human\n\nMany product formulas of the theory of partitions and of theta functions are special cases.\nIt is the starting point for the bilateral series ${}_r\\psi_r$, of which Bailey's ${}_6\\psi_6$ summation is the main generalization.\n",
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