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

It has been shown that the Cartan subalgebra of W∞-algebra is the space of the two-variable, definite-parity polynomials. Explicit expressions of these polynomials, and their basic properties are presented. It 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 Inonu–Wigner contraction and its inverse that form a bridge between the difference and differential calculus.