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Fungrim entry: 290f36

θ32 ⁣(0,τ)θ32 ⁣(0,2τ)=n=0r2 ⁣(2n+1)q2n+1   where q=eπiτ\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:τH\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
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
HHH\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))), Tuple(n, 0, Infinity)), Equal(q, Exp(Mul(Mul(ConstPi, ConstI), tau)))))),
    Variables(tau),
    Assumptions(Element(tau, HH)))

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2019-09-20 18:07:53.062439 UTC