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.
