A SIMPLER CHARACTERIZATION OF SHEFFER POLYNOMIALS

We characterize the Sheffer sequences by a single convolution identity F(y)p(n)(x) = SIGMA(k=0)n(p)k(x)p(n-k)(y), where F(y) is a shift-inva riant operator. We then study a generalization of the notion of Sheffe r sequences by removing the requirement that F(y) be shift-invariant. All these solutions can then be interpreted as cocommutative coalgebra s. We also show the connection with generalized translation operators as introduced by Delsarte. Finally, we apply the same convolution to s ymmetric functions where we find that the ''Sheffer'' sequences differ from ordinary full divided power sequences by only a constant factor.