Calogero-Sutherland-type quantum systems, generalized hypergeometric functions and superintegrability for integral chains

We reinvestigate the Calogero-Sutherland-type (CS-type) models and generalized hypergeometric functions. We construct the generalized CS operators for circular, Hermite, Laguerre, Jacobi and Bessel cases and establish the generalized Lassalle-Nekrasov correspondence. A family of operators are constructed based on the spherical degenerate double affine Hecke algebra. In terms of these operators, we provide concise representations and constraints for the generalized hypergeometric functions. We analyze the superintegrability for the $\beta$-deformed integrals, where the measures are associated with the corresponding ground state wave functions of Hermite, Laguerre, Jacobi and Bessel type CS models. Then based on the generalized Laplace transformation of Jack polynomials, we construct certain two integral chains and analyze the superintegrability property.