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

Series and product representations of Jacobi theta functions