Recurrences and explicit formulae for the expansion and connection coefficients in series of the product of two classical discrete orthogonal polynomials

Suppose that for an arbitrary function f(x,y) of two discrete variables, we have the formal expansions. [f(x,y)=sumlimits_{m,n=0}^{infty }a_{m,n},P_{m}(x)P_{n}(y),] ‎‎xmPj(x)=∑n=02mam,n(j)Pj+m−n(x)‎,‎we find the coefficients b(p,q,ℓ‎,‎r)i,j in the expansion‎ ‎‎xℓyr∇px∇qyf(x,y)=xℓ‎‎yrf(p,q)(x,y)=∑m,n=0∞‎‎a(p,q)m,nPm(x)Pn(y),a(0,0)m,n=am,n‎, ‎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.