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It has been shown that the Cartan subalgebra of $W_{\infty}$- algebra is thespace of the two-variable, definite-parity polynomials. Explicit expressions ofthese polynomials, and their basic properties are presented. Also has beenshown that they carry the infinite dimensional irreducible representation ofthe $su(1,1)$ algebra having the spectrum bounded from below. A realization ofthis algebra in terms of difference operators is also obtained. For particularvalues of the ordering parameter $s$ they are identified with the classicalorthogonal 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. Exactscattering states with zero energy for a family of potentials are expressed interms 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 betweenthe difference and differential calculus.Comment: 17 pages,no figure. to appear in J. Math.Phy

Ordered products, W∞-algebra, and two-variable, definite-parity, orthogonal polynomials