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 Anand Cnto the hyperbolic hypergeometric integrals, we apply the limit w1 -> -w2 for their quasiperiods (corresponding to b -> 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 Cn-root system related to the recently derived generalized complex Selberg integral. For some hyperbolic beta integrals we consider a special limit w1 -> w2 (or b -> 1) and obtain new hypergeometric identities for sums of integrals of rational functions. (c) 2024 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.