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Complete Monotonicity of classical theta functions and applications

We produce trigonometric expansions for Jacobi theta functions\\ $θ_j(u,τ), j=1,2,3,4$\ where $τ=iπt, t > 0$. This permits us to prove that\ $\log \frac{θ_j(u, t)}{θ_j(0, t)}, j=2,3,4$ and $\log \frac{θ_1(u, t)}{πθ'_1(0, t)}$ as well as $\frac{\frac{δθ_j}{δu}}{θ_j}$ as functions of $t$ are completely monotonic. We also interested in the quotients $S_j(u,v,t) = \frac{θ_j(u/2,iπt)}{θ_j(u/2,iπt)}$. For fixed $u,v$ such that $0\leq u < v < 1$ we prove that the functions $\frac{(\fracδ{δt}S_j)}{S_j}$ for $j=1,4$ as well as the functions $-\frac{(\fracδ{δt}S_j)}{S_j}$ for $j=2,3$ are completely monotonic for $t \in ]0,\infty[$.\\ {\it Key words and phrases} : theta functions, elliptic functions, complete monotonicity.