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Fungrim entry: 7c90eb

θ4k ⁣(0,τ)=n=0(1)nrk ⁣(n)qn   where q=eπiτ\theta_{4}^{k}\!\left(0, \tau\right) = \sum_{n=0}^{\infty} {\left(-1\right)}^{n} r_{k}\!\left(n\right) {q}^{n}\; \text{ where } q = {e}^{\pi i \tau}
Assumptions:kZ0andτHk \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H}
TeX:
\theta_{4}^{k}\!\left(0, \tau\right) = \sum_{n=0}^{\infty} {\left(-1\right)}^{n} r_{k}\!\left(n\right) {q}^{n}\; \text{ where } q = {e}^{\pi i \tau}

k \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H}
Definitions:
Fungrim symbol Notation Short description
Powab{a}^{b} Power
JacobiThetaθj ⁣(z,τ)\theta_{j}\!\left(z , \tau\right) Jacobi theta function
Sumnf ⁣(n)\sum_{n} f\!\left(n\right) Sum
SquaresRrk ⁣(n)r_{k}\!\left(n\right) Sum of squares function
Infinity\infty Positive infinity
Expez{e}^{z} Exponential function
ConstPiπ\pi The constant pi (3.14...)
ConstIii Imaginary unit
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
HHH\mathbb{H} Upper complex half-plane
Source code for this entry:
Entry(ID("7c90eb"),
    Formula(Equal(Pow(JacobiTheta(4, 0, tau), k), Where(Sum(Mul(Mul(Pow(-1, n), SquaresR(k, n)), Pow(q, n)), Tuple(n, 0, Infinity)), Equal(q, Exp(Mul(Mul(ConstPi, ConstI), tau)))))),
    Variables(k, tau),
    Assumptions(And(Element(k, ZZGreaterEqual(0)), Element(tau, HH))))

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Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-09-20 18:07:53.062439 UTC