Additive Matrix Convolutions of P\'olya Ensembles and Polynomial Ensembles

Recently subclasses of polynomial ensembles for additive and multiplicative matrix convolutions were identified which were called P\'olya ensembles (or polynomial ensembles of derivative type). Those ensembles are closed under the respective convolutions and, thus, build a semi-group when adding by hand a unit element. They even have a semi-group action on the polynomial ensembles. Moreover in several works transformations of the bi-orthogonal functions and kernels of a given polynomial ensemble were derived when performing an additive or multiplicative matrix convolution with particular P\'olya ensembles. For the multiplicative matrix convolution on the complex square matrices the transformations were even done for general P\'olya ensembles. In the present work we generalize these results to the additive convolution on Hermitian matrices, on Hermitian anti-symmetric matrices, on Hermitian anti-self-dual matrices and on rectangular complex matrices. For this purpose we derive the bi-orthogonal functions and the corresponding kernel for a general P\'olya ensemble which was not done before. With the help of these results we find transformation formulas for the convolution with a fixed matrix or a random matrix drawn from a general polynomial ensemble. As an example we consider P\'olya ensembles with an associated weight which is a P\'olya frequency function of infinite order. But we also explicitly evaluate the Gaussian unitary ensemble as well as the complex Laguerre (aka Wishart, Ginibre or chiral Gaussian unitary) ensemble. All results hold for finite matrix dimension. Furthermore we derive a recursive relation between Toeplitz determinants which appears as a by-product of our results.