Recently, subclasses of polynomial ensembles for additive and multiplicative matrix convolutions were identified which were called Pólya 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 Pólya ensembles. For the multiplicative matrix convolution on the complex square matrices the transformations were even done for general Pólya 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 Pólya 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 Pólya ensembles with an associated weight which is a Pólya 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.

中文翻译：

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Pólya 集成和多项式集成的加性矩阵卷积

最近，确定了用于加法和乘法矩阵卷积的多项式集成的子类，称为 Pólya 集成（或导数类型的多项式集成）。这些集合在各自的卷积下是封闭的，因此在手动添加单位元素时构建了一个半群。他们甚至对多项式集合有一个半群作用。此外，在一些工作中，当对特定的 Pólya 系综执行加法或乘法矩阵卷积时，导出了给定多项式系综的双正交函数和核的变换。对于复方矩阵上的乘法矩阵卷积，甚至对一般 Pólya 集成进行了转换。在目前的工作中，我们将这些结果推广到 Hermitian 矩阵上的加性卷积，Hermitian 反对称矩阵、Hermitian 反对自对偶矩阵和矩形复矩阵。为此，我们推导了以前没有做过的一般 Pólya 集成的双正交函数和相应的内核。在这些结果的帮助下，我们找到了具有固定矩阵或从一般多项式集合中提取的随机矩阵的卷积的变换公式。例如，我们考虑具有关联权重的 Pólya 系综，该权重是无限阶的 Pólya 频率函数。但我们也明确地评估了高斯酉系综以及复 Laguerre（又名 Wishart、Ginibre 或手性高斯酉）系综。所有结果都适用于有限矩阵维数。此外，