  The equilibrium positions of the multi-particle classical
Calogero-Sutherland-Moser (CSM) systems with rational/trigonometric potentials
associated with the classical root systems are described by the classical
orthogonal polynomials; the Hermite, Laguerre and Jacobi polynomials. The
eigenfunctions of the corresponding single-particle quantum CSM systems are
also expressed in terms of the same orthogonal polynomials. We show that this
interesting property is inherited by the Ruijsenaars-Schneider-van Diejen
(RSvD) systems, which are integrable deformation of the CSM systems; the
equilibrium positions of the multi-particle classical RSvD systems and the
eigenfunctions of the corresponding single-particle quantum RSvD systems are
described by the same orthogonal polynomials, the continuous Hahn (special
case), Wilson and Askey-Wilson polynomials. They belong to the Askey-scheme of
the basic hypergeometric orthogonal polynomials and are deformation of the
Hermite, Laguerre and Jacobi polynomials, respectively. The Hamiltonians of
these single-particle quantum mechanical systems have two remarkable
properties, factorization and shape invariance.
