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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 and to the hyperbolic hypergeometric integrals, we apply the limit for their quasiperiods (corresponding to 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 -root system related to the recently derived generalized complex Selberg integral. For some hyperbolic beta integrals we consider a special limit (or) and obtain new hypergeometric identities for sums of integrals of rational functions.