Probabilistic aspects of Jacobi theta functions

Authors

  • Paavo Salminen
  • Christophe Vignat

DOI:

https://doi.org/10.7146/math.scand.a-148416

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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Published

2024-11-04

How to Cite

Salminen, P., & Vignat, C. (2024). Probabilistic aspects of Jacobi theta functions. MATHEMATICA SCANDINAVICA, 130(3). https://doi.org/10.7146/math.scand.a-148416

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Articles