Eliminations in Weyl algebras and identities. (Q817239)

Consider two operators \(N,K\) on the algebra of polynomials in two variables \(n,k\), namely \[ N(f(n,k))=f(n+1,k),\quad K(f(n,k))=f(n,k+1). \] Denote by \(C\langle N,K,n,k\rangle\) the algebra of operators on polynomials generated by \(N,K,n,k\), where \(n,k\) are identified with operators of multiplication. Given \(P,Q\in C\langle N,K,n,k\rangle\) there exist nontrivial elements \(u,v,r,s\in C\langle N,K,n,k\rangle\) such that \(uP=vQ + R\), \(rP=sQ\) and \(\deg_kR<\deg_kQ\). This result is applied to elimination in Weyl algebras, for verification of termination of some hypergeometric identities, of identities in two variables, and so on.