In this paper, we study the generalized complete (p, q)-elliptic integrals of first and second kind as an application trigonometric functions with two parameters, establish monotonicity, convexity concavity these functions. particular, some Turán type inequalities are given. Finally, also show new series representations by applying Alzer Richard’s methods.

We study the elliptic Cn and BCn Ruijsenaars-Schneider models which is elliptic generalization of system given in [1]. The Lax pairs for these models are constructed by Hamiltonian reduction technology. We show that the spectral curves can be parameterized by the involutive integrals of motion for these models. Taking nonrelativistic limit and scaling limit, we verify that they lead to the syst...

We show that spin generalization of elliptic Calogero-Moser system, elliptic extension of Gaudin model and their cousins can be treated as a degenerations of Hitchin systems. Applications to the constructions of integrals of motion, angle-action variables and quantum systems are discussed. The constructions are motivated by the Conformal Field Theory, and their quantum counterpart can be treate...