Ordered Products, $W_{\infty}$-Algebra, and Two-Variable, Definite-Parity, Orthogonal Polynomials
It has been shown that the Cartan subalgebra of $W_{\infty}$- algebra is the space of the two-variable, definite-parity polynomials. Explicit expressions of these polynomials, and their basic properties are presented. Also has been shown that they carry the infinite dimensional irreducible representation of the $su(1,1)$ algebra having the spectrum bounded from below. A realization of this algebra in terms of difference operators is also obtained. For particular values of the ordering parameter $s$ they are identified with the classical orthogonal polynomials of a discrete variable, such as the Meixner, Meixner-Pollaczek, and Askey-Wilson polynomials. With respect to variable $s$ they satisfy a second order eigenvalue equation of hypergeometric type. Exact scattering 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 \.{I}n\"{o}n\"{u}-Wigner contraction and its inverse that form bridge between the difference and differential calculus.