Limits of Bessel functions for root systems as the rank tends to infinity - Brennecken, Dominik; Rösler, Margit (Indagationes Mathematicae)
We study the asymptotic behaviour of Bessel functions associated of root systems of type $A_{n-1}$ and type $B_n$ with positive multiplicities as the rank $n$ tends to infinity. In both cases, we characterize the possible limit functions and the Vershik-Kerov type sequences of spectral parameters for which such limits exist. In the type $A$ case, this gives a new and very natural approach to recent results by Assiotis and Najnudel in the context of $β$-ensembles in random matrix theory. These results generalize known facts about the approximation of the (positive-definite) Olshanski spherical functions of the space of infinite-dimensional Hermitian matrices over $\mathbb F = \mathbb R, \mathbb C, \mathbb H$ (with the action of the associated infinite unitary group) by spherical functions of finite-dimensional spaces of Hermitian matrices. In the type B case, our results include asymptotic results for the spherical functions associated with the Cartan motion groups of non-compact Grassmannians as the rank goes to infinity, and a classification of the Olshanski spherical functions of the associated inductive limits.

Hankel transform, K-Bessel functions and zeta distributions in the Dunkl setting - Brennecken, Dominik (Journal of Mathematical Analysis and Applications)
Keywords: Applied Mathematics; Analysis

The Laplace transform in Dunkl theory - Brennecken, Dominik; Rösler, Margit (Women in Analysis and PDE)

Dunkl convolution and elliptic regularity for Dunkl operators - Brennecken, Dominik (Mathematische Nachrichten)
Abstract We discuss in which cases the Dunkl convolution of distributions , possibly both with non‐compact support, can be defined and study its analytic properties. We prove results on the (singular‐)support of Dunkl convolutions. Based on this, we are able to prove a theorem on elliptic regularity for a certain class of Dunkl operators, called elliptic Dunkl operators. Finally, for the root system we consider the Riesz distributions and prove that their Dunkl convolution exists and that holds.

The Dunkl-Laplace transform and Macdonald’s hypergeometric series - Brennecken, Dominik; Rösler, Margit (Transactions of the American Mathematical Society)

Algebraically Independent Generators for the Algebra of Invariant Differential Operators on SLn(R)/SOn(R) - Brennecken, Dominik; Hilgert, Joachim; Ciardo, Lorenzo (Journal of Lie Theory)