Generalized Bessel function of Type D

Product formula for Jacobi polynomials, spherical harmonics and generalized Bessel function of dihedral type

We work out the expression of the generalized Bessel function of type B in the two-rank case. This is done using Dijskma and Koornwinder's product formula for Jacobi polynomials and the obtained expression is given by multiple integrals involving only a normalized modified Bessel function and two symmetric Beta distributions. We think of that expression as the major step toward the explicit expression of the Dunkl's intertwining V operator reflections-invariant functions. Finally, we give in the same setting an explicit formula for the action of V on a product of a power of the norm and a sphe…

Explicit solutions to fractional diffusion equations via Generalized Gamma Convolution

In this paper we deal with Mellin convolution of generalized Gamma densities which leads to integrals of modified Bessel functions of the second kind. Such convolutions allow us to explicitly write the solutions of the time-fractional diffusion equations involving the adjoint operators of a square Bessel process and a Bessel process.

广义Bessel-Dunkl扩散

Generalized Bessel-Dunkl diffusions

本文针对经典根系Weyl室中的Bessel-Dunkl型扩散，建立了包含时变、位变系数及可退化奇异排斥项的统一理论，证明了弱存在性、路径唯一性、强存在性及平均场收敛定理。

通过径向 Dunkl 过程的大偏差研究广义 Bessel 函数的渐近性与权重重数

Asymptotics of Generalized Bessel Functions and Weight Multiplicities via Large Deviations of Radial Dunkl Processes

本文基于径向 Dunkl 过程流体力学极限的大偏差原理，证明了广义 Bessel 函数的大 N 渐近变分公式及径向 Heckman-Opdam 过程的大偏差原理，并应用于确定大秩极限下单李代数不可约表示的权重重数渐近行为。

A power series identity and Bessel-type integrals over unitary groups

In 2008, Lehner, Wettig, Guhr and Wei conjectured a power series identity and showed that it implied a determinantal formula for a Bessel-type integral over the unitary supergroup. The integral is the supersymmetric extension of Bessel-type integrals over the unitary group appearing as partition functions in quantum chromodynamics. The identity is proved by interpreting both sides as the same unitary integral, which can be computed using the Cartan decomposition. An equivalent identity of Schur functions is also given and interpreted probabilistically.

Limit theorems for Bessel and Dunkl processes of large dimensions and free convolutions

We study Bessel and Dunkl processes $(X_{t,k})_{t\ge0}$ on $\mathbb R^N$ with possibly multivariate coupling constants $k\ge0$. These processes describe interacting particle systems of Calogero-Moser-Sutherland type with $N$ particles. For the root systems $A_{N-1}$ and $B_N$ these Bessel processes are related with $β$-Hermite and $β$-Laguerre ensembles. Moreover, for the frozen case $k=\infty$, these processes degenerate to deterministic or pure jump processes. We use the generators for Bessel and Dunkl processes of types A and B and derive analogues of Wigner's semicircle and Marchenko-Pastu…