Complex and rational hypergeometric functions on root systems

We consider some new limits for the elliptic hypergeometric integrals on root systems. After the degeneration of elliptic beta integrals of type I and type II for root systems $A_n$ and $C_n$ to the hyperbolic hypergeometric integrals, we apply the limit ${\omega }_1\to -{\omega }_2$ for their quasiperiods (corresponding to $b\to i$ in the two-dimensional conformal field theory) and obtain complex beta integrals in the Mellin–Barnes representation admitting exact evaluation. Considering type I elliptic hypergeometric integrals of a higher order obeying nontrivial symmetry transformations, we derive their descendants to the level of complex hypergeometric functions and prove the Derkachov–Manashov conjectures for functions emerging in the theory of non-compact spin chains. We describe also symmetry transformations for a type II complex hypergeometric function on the $C_n$-root system related to the recently derived generalized complex Selberg integral. For some hyperbolic beta integrals we consider a special limit ${\omega }_1\to {\omega }_2$ (or $b\to 1$) and obtain new hypergeometric identities for sums of integrals of rational functions.