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  "body": "\n## Definition\n\nLet $a, b \\in \\C \\setminus \\set{0}$ with $\\abs{ab} \u003c 1$.\n*Ramanujan's general theta function* is\n$$f(a, b) = \\sum_{n=-\\infty}^{\\infty} a^{n(n+1)/2}\\, b^{n(n-1)/2}.$$\nIts standard special cases, for $q \\in \\C$ with $\\abs{q} \u003c 1$, are\n$$\\varphi(q) = f(q, q), \\qquad \\psi(q) = f(q, q^3), \\qquad f(-q) = f(-q, -q^2).$$\n\n## Remarks\n\nThe series converges absolutely: for $n \\ge 0$ the term is $(ab)^{n(n-1)/2} a^n$, and for $n \u003c 0$ it is $(ab)^{m(m-1)/2} b^{m}$ with $m = -n$, so the terms decay like $\\abs{ab}^{n^2/2}$.\nReplacing $n$ by $-n$ shows $f(a, b) = f(b, a)$.\nThe [[Jacobi triple product|JFP2DGMNXT]] gives $f(a, b) = (-a; ab)_\\infty\\, (-b; ab)_\\infty\\, (ab; ab)_\\infty$ with the [[q-Pochhammer symbol|Y1X8ZZQ7VM]].\n\nExample: $\\psi(q) = \\sum_{n \\in \\Z} q^{n(n+1)/2 + 3n(n-1)/2} = \\sum_{n \\in \\Z} q^{2n^2 - n}$.\nThe exponents for $n = 0, 1, -1, 2, -2, \\dots$ are $0, 1, 3, 6, 10, \\dots$, the triangular numbers, each once, so $\\psi(q) = \\sum_{k \\ge 0} q^{k(k+1)/2}$.\nAlso $f(-q) = \\sum_{n \\in \\Z} (-1)^n q^{n(3n-1)/2} = (q; q)_\\infty$, by [[Euler's pentagonal number theorem|R587SR9GNV]].\n\nNon-example: $f(a, b)$ is not defined for $a = b = 1$, since $\\abs{ab} = 1$ and the series $\\sum_{n \\in \\Z} 1$ diverges.\n\nIn the notation of the Jacobi theta function, $f(q e^{2iz}, q e^{-2iz}) = \\vartheta_3(z, q)$.\n",
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