# Tagged Questions

For questions about $\theta$ functions (special functions of several complex variables).

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### Convexity of $\theta(q)$

Define Jacobi's (fourth) theta function with argument zero and nome $q$: $$\theta(q) = 1+2\sum_{n=1}^\infty (-1)^n q^{n^2}$$ plot of the function via Wolfram|Alpha plot of the function via Sage I ...
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### How to estimate a specific infinite sum

Let $M$ be an $n$ by $n$ matrix with each diagonal element equal to $k$ and each non-diagonal element equal to $k-1$ where $n$ and $k$ are positive integers. Let $k < n$ and we can assume both $k$ ...
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### About the sums $\sum_{n=1}^\infty x^{n^2}$ and $\sum_{n=1}^\infty \frac{x^n}{1+x^{2n}}$

Despite all my efforts trying to crack these, i haven't been able to do so. An approach that i've tried gives me somewhat of an asymptotic approximation, but still fails to produce the values near x=0....
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### Modular forms on the theta group

The theta group $\Gamma$ is the subgroup of $SL(2;\mathbb{Z})$ generated by $T=\begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}$ and $S=\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}.$ It ...
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### Implementation of Jacobi theta functions in Matlab

I need Jacobi theta functions for my Matlab program. The functions are not included in the predefined Matlab functions. Doing a simple Google search, I found a package developed by Moiseev I. in 2008. ...
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### Fibonacci-related infinite sum

Prove that $$\sum_{n=0}^\infty \frac{1}{F_{2n+1}}=\frac{\sqrt{5}}{4}\theta_2^2\bigg(\frac{3-\sqrt{5}}{2}\bigg)$$ and $$\sum_{n=0}^\infty \frac{1}{F_{2n+1}+F_{2k-1}}=\frac{(2k-1)\sqrt{5}}{2F_{2k-1}}$$ ...
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### Equation of the curve corresponding to a principal polarization

Let $\mathbb{C}^2/\Lambda$ be a principally polarized abelian surface. I think it is well-known how to write down the equation of the divisor (Riemann surface) corresponding to the polarization, in ...
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### Does this function appear in theory of modular forms?

Does the following function $\phi(q,z)$ appear in theory of modular forms $$\phi(q,z):=\prod_{n=-\infty}^{\infty}(1+q^nz)$$ for $(q,z)$ in some domain in $\mathbb{C}^2$? Any reference will be ...
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### q-theta function and their properties

I want to compute the residue integral for the $q$-theta function, and derive its properties. First, I'll briefly explain the definition \begin{align} & \theta(a;q)=(a;q)(q/a;q)=\prod_{i=0}^{\...
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### Rapidly convergent series for $\sum_{J=0}^{\infty} (2 J + 1) e^{-\beta J(J+1)}$ (rigid rotor)

I need to evaluate this series for arbitrary $\beta > 0$: $Q = \sum_{J=0}^{\infty} (2 J + 1) e^{-\beta J(J+1)}$ Is it related to a known transcendental function? From the research I did, it ...
### Let $\theta(z) = \sum q^{n^2}$, is $\theta(-1/z)$ also a theta function?
I am learning about theta functions. Let $q = e^{2\pi i \, z}$ and $\theta(z) = \sum q^{n^2}$. How does it behave under $\mathrm{SL}_2(\mathbb{Z})$ ? In general we have:  \theta\left( - \frac{1}...