A Limit Relation for Dunkl-Bessel Functions of Type A and B

TL;DR

This paper establishes a limit relation connecting Dunkl-Bessel functions of types A and B, showing how type B functions approximate type A functions under certain parameter limits and scalings.

Contribution

It introduces a new limit relation for Dunkl-Bessel functions of type B, linking them to type A functions, with improved estimates for specific parameters.

Findings

Dunkl-Bessel functions of type B approximate type A functions as a parameter tends to infinity.

A refined estimate is provided for certain values of the multiplicity parameter $k_2$.

The limit relation enhances understanding of the asymptotic behavior of Dunkl-Bessel functions.

Abstract

We prove a limit relation for the Dunkl-Bessel function of type $B_N$ with multiplicity parameters $k_1$ on the roots $\pm e_i$ and $k_2$ on $\pm e_i\pm e_j$ where $k_1$ tends to infinity and the arguments are suitably scaled. It gives a good approximation in terms of the Dunkl-type Bessel function of type $A_{N-1}$ with multiplicity $k_2$. For certain values of $k_2$ an improved estimate is obtained from a corresponding limit relation for Bessel functions on matrix cones.