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arXiv · 2303.05942

Probabilistic aspects of Jacobi theta functions

Paavo Salminen·Christophe Vignat
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Abstract

In this note we deduce well known modular identities for Jacobi theta functions using the spectral representations associated with the real valued Brownian motion taking values on $[-1,+1]$. We consider two cases: (i) reflection at $-1$ and $+1$, (ii) killing at $-1$ and $+1$. It is seen that these two representations give, in a sense, most compact forms of the modular theta-function identities. We study also discrete Gaussian distributions generated by theta functions, and derive, in particular, addition formulas for discrete Gaussian variables.

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Paavo Salminen, Christophe Vignat. 2023-03-10. Probabilistic aspects of Jacobi theta functions. https://arxiv.org/abs/2303.05942

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