RECURRENCES AND EXPLICIT FORMULAE FOR THE EXPANSION AND CONNECTION COEFFICIENTS IN SERIES OF THE PRODUCT OF TWO CLASSICAL DISCRETE ORTHOGONAL POLYNOMIALS

Abstract

Suppose that for an arbitrary function f(x, y) of two discrete variables, we have the formal expansions.

f(x, y) = Sigma(infinity )(m,n=0)a(m,n )P(m) (x) P-n (y),

del(p)(x)del(q)(y)f(x,y) = f((p,q)) (x,y) = Sigma(infinity)(m,n=0) a(m,n)((p,q)) P-m(x) P-n(y), a(m,n)((0,0)) = a(m,n),

where P-n (x), n = 0,1, 2, . . . are the Hahn, Meixner, Kravchuk and Charlier polynomials.

We prove formulae which give a(m,n)((p,q)), as a linear combination of a(i,j), i, j = 0, 1, 2, . . . . Using the moments of a discrete orthogonal polynomial,

x(m) P-j(x) = Sigma(2m)(n=0) a(m,n )(j) Pj+m-n (x),

we find the coefficients b(i,j)((p,q,l,r)) in the expansion

x(l) y(r) del(p)(x)del(q)(y) f(x,y) = x(l) y(r) f((p,q)) (x,y) = Sigma(infinity)(i,j=0) b(i,j)((p,q,l,r)) P-i(x) P-j(y).

We give applications of these results in solving partial difference equations with varying polynomial coefficients, by reducing them to recurrence relations (difference equations) in the expansion coefficients of the solution.