Some freezing limits for Bessel functions and Bessel processes with drift of type BN
Abstract
We use the series representation of the Bessel functions of type BN in terms of Jack polynomials and show that limk1→∞ J(k1,k2)B(x,k1y)1/k1= 2N∏i=1N ( e√(1+xi2 yi2)−1· 1/(√(1+xi2 yi2)+1) ) for x,y∈ℝN and k2≥0. Moreover, the known Laplace-type integral representations for N≥1 and k2=0,1/2,1,2 and for N=2 and k2>0 by Rösler and Demni respectively lead to related limits for limk1 → ∞ ∂xj J(k1,k2)B(x,k1y) /(k1 · J(k1,k2)B(x,k1 y)) (j=1,…,N). These limits lead to weak limit results for the associated Bessel processes with drift. For k2=1/2,1,2, these limit results have applications to radial parts of Brownian motions with drift on the M× N-dimensional matrices over ℝ,ℂ, and the quaternions for M→∞. We also discuss these limits in the Dunkl case N=1.
Related
- A central limit theorem for Bessel and Dunkl processes with drift
- On the simultaneous zeroes of Bessel processes
- A relationship between three-dimensional Bessel processes in terms of Pitman's transformation
- Indefinite integrals of Bessel and Struve functions with half-integer indices, and incomplete gamma functions with integer indices, all multiplied by Exp(−ax2) and divided by powers
- Exact barycentric rational interpolation of analytic functions with an application to the Bessel functions
BibTeX
@misc{richter2026some,
title = {{Some freezing limits for Bessel functions and Bessel processes with drift of type $B_N$}},
author = {Jan Richter and Michael Voit},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.21071},
eprint = {2609.21071},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.21071},
}
RIS
TY - UNPB
TI - Some freezing limits for Bessel functions and Bessel processes with drift of type B_N
AU - Jan Richter
AU - Michael Voit
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.21071
UR - https://arxiv.org/abs/2609.21071
AB - We use the series representation of the Bessel functions of type B_N in terms of Jack polynomials and show that lim_k_1→∞ J_(k_1,k_2)^B(x,k_1y)^1/k_1= 2^N∏_i=1^N ( e^√(1+x_i^2 y_i^2)−1· 1/(√(1+x_i^2 y_i^2)+1) ) for x,y∈ℝ^N and k_2≥0. Moreover, the known Laplace-type integral representations for N≥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 lim_k_1 → ∞ ∂_x_j J_(k_1,k_2)^B(x,k_1y) /(k_1 · J_(k_1,k_2)^B(x,k_1 y)) (j=1,…,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× N-dimensional matrices over ℝ,ℂ, and the quaternions for M→∞. We also discuss these limits in the Dunkl case N=1.
ER -