On the Symmetrized Kronecker Power of a Matrix and Extensions of Mehler’s Formula for Hermite Polynomials

Mehler’s formula expresses the exponential of a quadratic form in two variables as a series of products of Hermite polynomials. We give several useful generalizations of this formula to the case of n variables, being guided in this work by interpretations in terms of Gaussian variates. Along the way we encounter the symmetrized Kronecker power of a matrix and we present a new generating function and recipe for calculating this quantity.