An exact formula for partition functions in 3d field theories was recently suggested by Jafferis, and Hama, Hosomichi, and Lee. These functions are expressed in terms of specific $q$-hypergeometric integrals whose key building block is the double sin
e function (or the hyperbolic gamma function). Elliptic hypergeometric integrals, discovered by the second author, define 4d superconformal indices. Using their reduction to the hyperbolic level, we describe a general scheme of reducing 4d superconformal indices to 3d partition functions which imply an efficient way of getting 3d $mathcal{N}=2$ supersymmetric dualities for both SYM and CS theories from the parent 4d $mathcal{N}=1$ dualities for SYM theories. As an example, we consider explicitly the duality pattern for 3d $mathcal{N}=2$ SYM and CS theories with SP(2N) gauge group with the antisymmetric tensor matter.
