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  "body": "\n## Statement\n\nSrinivasa Ramanujan gave the identity\n\n$$\\;_1\\psi_1 \\left[\\begin{matrix} a \\\\ b \\end{matrix} ; q,z \\right] = \\sum_{n=-\\infty}^\\infty \\frac {(a;q)_n} {(b;q)_n} z^n = \\frac {(b/a,q,q/az,az;q)_\\infty } {(b,b/az,q/a,z;q)_\\infty}$$\n\nvalid for |*q*| \\\u003c 1 and |*b*/*a*| \\\u003c |*z*| \\\u003c 1. Similar identities for $\\;_6\\psi_6$ have been given by Bailey. Such identities can be understood to be generalizations of the Jacobi triple product theorem, which can be written using q-series as\n\n$$\\sum_{n=-\\infty}^\\infty q^{n(n+1)/2}z^n = (q;q)_\\infty \\; (-1/z;q)_\\infty \\; (-zq;q)_\\infty.$$\n\nGwynneth Coogan and Ken Ono give a related [[formal power series|033WGN00BF]]\n\n$$A(z;q) \\stackrel{\\rm{def}}{=} \\frac{1}{1+z} \\sum_{n=0}^\\infty \\frac{(z;q)_n}{(-zq;q)_n}z^n = \\sum_{n=0}^\\infty (-1)^n z^{2n} q^{n^2}.$$\n",
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