# Fungrim entry: a5e568

$\theta_{3}^{k}\!\left(0, \tau\right) = \sum_{n=0}^{\infty} r_{k}\!\left(n\right) {q}^{n}\; \text{ where } q = {e}^{\pi i \tau}$
Assumptions:$k \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H}$
TeX:
\theta_{3}^{k}\!\left(0, \tau\right) = \sum_{n=0}^{\infty} 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
Pow${a}^{b}$ Power
JacobiTheta$\theta_{j}\!\left(z , \tau\right)$ Jacobi theta function
Sum$\sum_{n} f(n)$ Sum
SquaresR$r_{k}\!\left(n\right)$ Sum of squares function
Infinity$\infty$ Positive infinity
Exp${e}^{z}$ Exponential function
Pi$\pi$ The constant pi (3.14...)
ConstI$i$ Imaginary unit
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
HH$\mathbb{H}$ Upper complex half-plane
Source code for this entry:
Entry(ID("a5e568"),
Formula(Equal(Pow(JacobiTheta(3, 0, tau), k), Where(Sum(Mul(SquaresR(k, n), Pow(q, n)), For(n, 0, Infinity)), Equal(q, Exp(Mul(Mul(Pi, ConstI), tau)))))),
Variables(k, tau),
Assumptions(And(Element(k, ZZGreaterEqual(0)), Element(tau, HH))))

## Topics using this entry

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

2020-01-31 18:09:28.494564 UTC