  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.
