Complete Monotonicity of classical theta functions and applications

TL;DR

This paper establishes the complete monotonicity of various classical theta functions and their ratios with respect to the parameter t, providing new insights into their analytical properties and potential applications.

Contribution

The paper introduces trigonometric expansions for Jacobi theta functions and proves their complete monotonicity in t, a novel analytical property not previously established.

Findings

Proved complete monotonicity of log ratios of theta functions in t.

Established monotonicity properties of quotients involving theta functions.

Provided new expansions for Jacobi theta functions.

Abstract

We produce trigonometric expansions for Jacobi theta functions\\ $\theta_j(u,\tau), j=1,2,3,4$\ where $\tau=i\pi t, t > 0$. This permits us to prove that\ $\log \frac{\theta_j(u, t)}{\theta_j(0, t)}, j=2,3,4$ and $\log \frac{\theta_1(u, t)}{\pi \theta'_1(0, t)}$ as well as $\frac{\frac{\delta\theta_j}{\delta u}}{\theta_j}$ as functions of $t$ are completely monotonic. We also interested in the quotients $S_j(u,v,t) = \frac{\theta_j(u/2,i\pi t)}{\theta_j(u/2,i\pi t)}$. For fixed $u,v$ such that $0\leq u < v < 1$ we prove that the functions $\frac{(\frac{\delta}{\delta t}S_j)}{S_j}$ for $j=1,4$ as well as the functions $-\frac{(\frac{\delta}{\delta t}S_j)}{S_j}$ for $j=2,3$ are completely monotonic for $t \in ]0,\infty[$.\\ {\it Key words and phrases} : theta functions, elliptic functions, complete monotonicity.