ORDERED PRODUCTS, W-INFINITY-ALGEBRA, AND 2-VARIABLE, DEFINITE-PARITY, ORTHOGONAL POLYNOMIALS

It has been shown that the Cartan subalgebra of W-infinity-algebra is the space of the two-variable, definite-parity polynomials. Explicit e xpressions of these polynomials, and their basic properties are presen ted. It also has been shown that they carry the infinite dimensional i rreducible representation of the su(1, 1) algebra having the spectrum bounded from below. A realization of this algebra in terms of differen ce operators is also obtained. For particular values of the ordering p arameter s they are identified with the classical orthogonal polynomia ls of a discrete variable, such as the Meixner, Meixner-Pollaczek, and Askey-Wilson polynomials. With respect to variable s they satisfy a s econd order eigenvalue equation of hypergeometric type. Exact scatteri ng states with zero energy for a family of potentials are expressed in terms of these polynomials. It has been put forward that it is the In onu-Wigner contraction and its inverse that form a bridge between the difference and differential calculus. (C) 1998 American Institute of P hysics. [S0022-2488(98)00304-1].