Complete Monotonicity of classical theta functions and applications

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 $\log \frac{{\theta }_1(u,t)}{\pi {\theta }_1^{\prime }(0,t)}$ 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 $0\le u<v<1$ 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.

19 pages