Fungrim entry: 290f36

\theta_{3}^{2}\!\left(0, \tau\right) - \theta_{3}^{2}\!\left(0, 2 \tau\right) = \sum_{n=0}^{\infty} r_{2}\!\left(2 n + 1\right) {q}^{2 n + 1}\; \text{ where } q = {e}^{\pi i \tau}

Assumptions:

\tau \in \mathbb{H}

TeX:

\theta_{3}^{2}\!\left(0, \tau\right) - \theta_{3}^{2}\!\left(0, 2 \tau\right) = \sum_{n=0}^{\infty} r_{2}\!\left(2 n + 1\right) {q}^{2 n + 1}\; \text{ where } q = {e}^{\pi i \tau}

\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
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("290f36"),
    Formula(Equal(Sub(Pow(JacobiTheta(3, 0, tau), 2), Pow(JacobiTheta(3, 0, Mul(2, tau)), 2)), Where(Sum(Mul(SquaresR(2, Add(Mul(2, n), 1)), Pow(q, Add(Mul(2, n), 1))), For(n, 0, Infinity)), Equal(q, Exp(Mul(Mul(Pi, ConstI), tau)))))),
    Variables(tau),
    Assumptions(Element(tau, HH)))

Topics using this entry

Series and product representations of Jacobi theta functions