Some freezing limits for Bessel functions and Bessel processes with drift of type $B_N$

Abstract

We use the series representation of the Bessel functions of type $B_N$ in terms of Jack polynomials and show that $$\underset{k_1\to \infty }{\lim }{J}_{(k_1,k_2)}^B(x,k_{1}y{)}^{1/k_1}=2^N\prod _{i=1}^N(e^{\sqrt{1+x_i^2{y}_i^2}-1}\cdot \frac{1}{\sqrt{1+x_i^2{y}_i^2}+1})$$ for $x,y\in {\mathbb{R}}^N$ and $k_2\ge 0$. Moreover, the known Laplace-type integral representations for $N\ge 1$ and $k_2=0,1/2,1,2$ and for $N=2$ and $k_2>0$ by Rösler and Demni respectively lead to related limits for $$\underset{k_1\to \infty }{\lim }{\partial }_{x_j}{J}_{(k_1,k_2)}^B(x,k_{1}y)/(k_1\cdot J_{(k_1,k_2)}^B(x,k_{1}y))(j=1,\dots ,N).$$ These limits lead to weak limit results for the associated Bessel processes with drift. For $k_2=1/2,1,2$, these limit results have applications to radial parts of Brownian motions with drift on the $M\times N$-dimensional matrices over $\mathbb{R},\mathbb{C}$, and the quaternions for $M\to \infty$. We also discuss these limits in the Dunkl case $N=1$.