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),]
ýýxmPj(x)=∑n=02mam,n(j)Pj−n(x)ý,
ýwe find the coefficients b(p,q,ℓý,ýr)i,j in the expansioný
ýýxℓyr∇px∇qyf(x,y)=xℓýýyrf(p,q)(x,y)=∑m,n=0∞ýýa(p,q)m,nPm(x)Pn(y),a(0,0)m,n=am,ný,
ýWe give applications of these results in solving partial differenceý ýequations with varying polynomial coefficientsý, ýby reducing them toý ýrecurrence relations (difference equations) in the expansioný ýcoefficients of the solutioný.