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

Source abstract

We use the series representation of the Bessel functions of type $B_N$ in terms of Jack polynomials and show that $$\lim_{k_1\to\infty} J_{(k_1,k_2)}^B(x,k_1y)^{1/k_1}= 2^N\prod\limits_{i=1}^N \Bigg( e^{\sqrt{1+x_i^2 y_i^2}-1}\cdot \frac{1}{\sqrt{1+x_i^2 y_i^2}+1} \Bigg)$$ for $x,y\in\mathbb R^N$ and $k_2\ge0$. Moreover, the known Laplace-type integral representations for $N\ge1$ 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 $$\lim_{k_1 \to \infty} \partial_{x_j} J_{(k_1,k_2)}^B(x,k_1y) /(k_1 \cdot J_{(k_1,k_2)}^B(x,k_1 y)) \quad (j=1,\ldots,N).$$ 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 $N=1$.