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Some explicit values for ratios of Theta-functions

In his notebooks [9], Ramanujan recorded several values of thetafunctions. B. C. Berndt and L-C. Zhang [6], Berndt and H. H. Chan [5] have proved all these evaluations. The main purpose of this paper is to establish several new explicit evaluations of ratios of thetafunctions. We also explicitly determinea(e 2 p 2 ),a(e 2 p 2=9 ),a(e p 2