Kronecker Product Identities from D-finite Symmetric Functions

Using an algorithm for computing the symmetric function Kronecker product of D-finite symmetric functions we find some new Kronecker product identities. The identities give closed form formulas for trace-like values of the Kronecker product. Introduction In the process of showing how the scalar product of symmetric functions can be used for enumeration purposes, Gessel [3], proved that this product, and the Kronecker product, preserve D-finiteness. Roughly, this means that if F and G are symmetric functions which both satisfy a particular kind of system of linear differential equations, then so will the scalar and Kronecker products of these functions. In an earlier work [2], we give algorithms to calculate both of these systems of differential equations. In this short note we use this algorithm in a symbolic way to find explicit expressions for Kronecker products of pairs of several common series of symmetric functions, such as complete (H = ∑ n hn), elementary (E = ∑ n en) and Schur (S = ∑ n ∑ λ`n sλ). Proposition 12 of [2], is the following identity, (∑