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

Radial Dunkl processes associated with Dihedral systems

We stduy radial Dunkl processes associated with dihedral systems: we derive the semi group, the generalized Bessel function, the Dunkl-Hermite polynomials. Then we give a skew product decomposition by means of independent Bessel processes and we compute the tail distribution of the first hitting time of the boundary of Weyl chamber.

Two Bessel Bridges Conditioned Never to Collide, Double Dirichlet Series, and Jacobi Theta Function

It is known that the moments of the maximum value of a one-dimensional conditional Brownian motion, the three-dimensional Bessel bridge with duration 1 started from the origin, are expressed using the Riemann zeta function. We consider a system of two Bessel bridges, in which noncolliding condition is imposed. We show that the moments of the maximum value is then expressed using the double Dirichlet series, or using the integrals of products of the Jacobi theta functions and its derivatives. Since the present system will be provided as a diffusion scaling limit of a version of vicious walker m…

Hard-edge asymptotics of the Jacobi growth process

We introduce a two parameter ($α, β>-1$) family of interacting particle systems with determinantal correlation kernels expressible in terms of Jacobi polynomials $\{ P^{(α, β)}_k \}_{k \geq 0}$. The family includes previously discovered Plancherel measures for the infinite-dimensional orthogonal and symplectic groups. The construction uses certain BC-type orthogonal polynomials which generalize the characters of these groups. The local asymptotics near the hard edge where one expects distinguishing behavior yields the multi-time $(α, β)$-dependent discrete Jacobi kernel and the multi-time $β$-…

Probability laws related to the Jacobi theta and Riemann zeta function and Brownian excursions

This paper reviews known results which connect Riemann's integral representations of his zeta function, involving Jacobi's theta function and its derivatives, to some particular probability laws governing sums of independent exponential variables. These laws are related to one-dimensional Brownian motion and to higher dimensional Bessel processes. We present some characterizations of these probability laws, and some approximations of Riemann's zeta function which are related to these laws.

Jacobi 系综随机矩阵的极端特征值

Extreme eigenvalues of random matrices from Jacobi ensembles

本文研究 Jacobi β-Ensembles 大矩阵的极端特征值分布，在 β 任意、α₁ 为整数时给出最小特征值的两项渐近公式，并推广至最大特征值及特殊情形。

关于冻结Calogero-Moser-Sutherland粒子模型的微分方程

On the differential equations of frozen Calogero-Moser-Sutherland particle models

本文研究冻结Calogero-Moser-Sutherland粒子模型的微分方程，证明相关常微分方程的特解吸引所有解，建立其与逆热方程的联系，并用于计算Bessel和Jacobi过程行列式公式的期望。