Expansions of the Jacobi Theta Functions and Complete Monotonicity Properties
Résumé
In this paper we provide trigonometric expansions and innite products for the theta functions θ j (u, τ), j = 1, 2, 3, 4. This allows us to highlight some complete monotonicity properties for log θ j (u,iπt) θ j (0,iπt) , j = 2, 3, 4 and log θ 1 (u,iπt) θ 1 (0,iπt) as functions with respect to t > 0. We also interested in the quotients S j (u, v, t) = θ j (u 2 ,iπt) θ j (u 2 ,iπt). Some others functions related to θ j will be considered.
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