Interlacing of positive real zeros of Bessel functions

Refined interlacing properties for zeros of paraorthogonal polynomials on the unit circle

The purpose of this note is to extend in a simple and unified way the known results on interlacing of zeros of paraorthogonal polynomials on the unit circle. These polynomials can be regarded as the characteristic polynomials of any matrix similar to an unitary upper Hessenberg matrix with positive subdiagonal elements.

Bounds for Extreme Zeros of Quasi-orthogonal Ultraspherical Polynomials

We discuss and compare upper and lower bounds obtained by two different methods for the positive zero of the ultraspherical polynomial $C_{n}^{(λ)}$ that is greater than $1$ when $-3/2 < λ< -1/2.$ Our first approach uses mixed three term recurrence relations and interlacing of zeros while the second approach uses a method going back to Euler and Rayleigh and already applied to Bessel functions and Laguerre and $q$-Laguerre polynomials. We use the bounds obtained by the second method to simplify the proof of the interlacing of the zeros of $(1-x^2)C_{n}^{(λ)}$ and $C_{n+1}^{(λ)}$, for $-3/2 < λ…

Products of Bessel and modified Bessel functions

The reality of the zeros of the product and cross-product of Bessel and modified Bessel functions of the first kind is studied. As a consequence the reality of the zeros of two hypergeometric polynomials is obtained together with the number of the Fourier critical points of the normalized forms of the product and cross-product of Bessel functions. Moreover, the interlacing properties of the real zeros of these products of Bessel functions and their derivatives are also obtained. As an application some geometric properties of the normalized forms of the cross-product and product of Bessel and m…

A function Class of strictly positive definite and logarithmically completely monotonic functions related to the modified Bessel functions

In this paper we give some conditions for a class of functions related to Bessel functions to be positive definite or strictly positive definite . We present some properties and relationships involving logarithmically completely monotonic functions and strictly positive definite functions. In particular, we are interested with the modified Bessel functions.

Angelesco-Jacobi多项式零点的交错性与单调性

Interlacing and monotonicity of zeros of Angelesco-Jacobi polynomials

研究Angelesco-Jacobi多项式零点随参数变化的交错性与单调性，扩展了已有结果并推广至相关多项式族。

Interlacing Log-concavity of the Boros-Moll Polynomials

We introduce the notion of interlacing log-concavity of a polynomial sequence $\{P_m(x)\}_{m\geq 0}$, where $P_m(x)$ is a polynomial of degree m with positive coefficients $a_{i}(m)$. This sequence of polynomials is said to be interlacing log-concave if the ratios of consecutive coefficients of $P_m(x)$ interlace the ratios of consecutive coefficients of $P_{m+1}(x)$ for any $m\geq 0$. Interlacing log-concavity is stronger than the log-concavity. We show that the Boros-Moll polynomials are interlacing log-concave. Furthermore we give a sufficient condition for interlacing log-concavity which i…

On the $ν$-zeros of the modified Bessel function $K_{iν}(x)$ of positive argument

The modified Bessel function of the second kind $K_{iν}(x)$ of imaginary order for fixed $x>0$ possesses a countably infinite sequence of real zeros. Recently it has been shown that the $n$th zero behaves like $ν_n\sim πn/\log\,n$ as $n\to\infty$. In this note we determine a more precise estimate for the bahaviour of these zeros for large $n$ by making use of the known asymptotic expansion of $K_{iν}(x)$ for large $ν$. Numerical results are presented to illustrate the accuracy of the expansion obtained.

The radius of convexity of normalized Bessel functions of the first kind

In this paper we determine the radius of convexity for three kind of normalized Bessel functions of the first kind. In the mentioned cases the normalized Bessel functions are starlike-univalent and convex-univalent, respectively, on the determined disks. The key tools in the proofs of the main results are some new Mittag-Leffler expansions for quotients of Bessel functions of the first kind, special properties of the zeros of Bessel functions of the first kind and their derivative, and the fact that the smallest positive zeros of some Dini functions are less than the first positive zero of the…

Bounds for radii of starlikeness of some $q$-Bessel functions

In this paper the radii of starlikeness of the Jackson and Hahn-Exton $q$-Bessel functions are considered and for each of them three different normalization are applied. By applying Euler-Rayleigh inequalities for the first positive zeros of these functions tight lower and upper bounds for the radii of starlikeness of these functions are obtained. The Laguerre-Pólya class of real entire functions plays an important role in this study. In particular, we obtain some new bounds for the first positive zero of the derivative of the classical Bessel function of the first kind.

Derivatives with respect to the order of the Bessel function of the first kind

An explicit expression of the k-th derivative of the Bessel function $J_ν(z)$, with respect to its order $ν$, is given. Particularizations for the cases of positive or negative $ν$ are considered.

High-order derivatives of the Bessel functions with an application

We determine the asymptotic behaviour of the $n$th derivatives of the Bessel functions $J_ν(a)$ and $K_ν(a)$, where $a$ is a fixed positive quantity, as $n\to\infty$. These results are applied to the asymptotic evaluation of two incomplete Laplace transforms of these Bessel functions on the interval $[0,a]$ as the transform variable $x\to+\infty$. Similar evaluation of the integrals involving the Bessel functions $Y_ν(t)$ and $I_ν(t)$ is briefly mentioned.

On the order derivatives of Bessel functions

The derivatives with respect to order ν for the Bessel functions of argument x (real or complex) are studied. Representations are derived in terms of integrals that involve the products pairs of Bessel functions, and in turn series expansions are obtained for these integrals. From the new integral representations, asymptotic approximations involving Airy functions are constructed for the order derivatives, for ν large and uniformly valid for unbounded positive real x.

An asymptotic expansion for a sum of modified Bessel functions with quadratic argument

We examine the sum of modified Bessel functions with argument depending quadratically on the summation index given by \[S_ν(a)=\sum_{n\geq 1} (\frac{1}{2} an^2)^{-ν} K_ν(an^2)\qquad (|\arg\,a|<π/2)\] as the parameter $|a|\to 0$. It is shown that the positive real $a$-axis is a Stokes line, where an infinite number of increasingly subdominant exponentially small terms present in the asymptotic expansion undergo a smooth, but rapid, transition as this ray is crossed. Particular attention is devoted to the details of the expansion on the Stokes line as $a\to 0$ through positive values. Numerical …

On a nonorthogonal polynomial sequence associated with Bessel operator

By means of the Bessel operator a polynomial sequence is constructed to which several properties are given. Among them, its explicit expression, the connection with the Euler numbers, its integral representation via the Kontorovich-Lebedev transform. Despite its non-orthogonality, it is possible to associate to the canonical element of its dual sequence a positive-definite measure as long as certain stronger constraints are imposed.

The evaluation of infinite sums of products of Bessel functions

We examine convergent representations for the sum of Bessel functions \[\sum_{n=1}^\infty \frac{J_μ(na) J_ν(nb)}{n^α}\] for $μ$, $ν\geq0$ and positive values of $a$ and $b$. Such representations enable easy computation of the series in the limit $a, b\to0+$. Particular attention is given to logarithmic cases that occur both when $a=b$ and $a\neq b$ for certain values of $α$, $μ$ and $ν$. The series when the first Bessel function is replaced by the modified Bessel function $K_μ(na)$ is also investigated, as well as the series with two modified Bessel functions.

On Basic Fourier-Bessel Expansions

When dealing with Fourier expansions using the third Jackson (also known as Hahn-Exton) $q$-Bessel function, the corresponding positive zeros $j_{kν}$ and the "shifted" zeros, $qj_{kν}$, among others, play an essential role. Mixing classical analysis with $q$-analysis we were able to prove asymptotic relations between those zeros and the "shifted" ones, as well as the asymptotic behavior of the third Jackson $q$-Bessel function when computed on the "shifted" zeros. A version of a $q$-analogue of the Riemann-Lebesgue theorem within the scope of basic Fourier-Bessel expansions is also exhibited.

On the real-rootedness of the local $h$-polynomials of edgewise subdivisions

Athanasiadis conjectured that, for every positive integer $r$, the local $h$-polynomial of the $r$th edgewise subdivision of any simplex has only real zeros. In this paper, based on the theory of interlacing polynomials, we prove that a family of polynomials related to the desired local $h$-polynomial is interlacing and hence confirm Athanasiadis' conjecture.

Asymptotics of some integrals involving modified Bessel and hyper-Bessel functions

We investigate the asymptotic expansion of integrals analogous to Ball's integral \[\int_0^\infty \left(\frac{Γ(1+ν)|J_ν(x)|}{(x/2)^ν}\right)^{\!n}dx\] for large $n$ in which the Bessel function $J_ν(x)$ is replaced by the modified Bessel functions $I_ν(x)$ and $K_ν(x)$ together with appropriate exponential factors $e^{\mp x}$, respectively. The above integral with $J_ν(x)$ replaced by a hyper-Bessel function of the type recently discussed in Aktas {\it et al.} [The Ramanujan J., 2019] and taken over a finite interval determined by the first positive zero of the function is also considered for…

贝塞尔函数零点的一致渐近展开

Uniform asymptotic expansions for the zeros of Bessel functions

本研究针对含大参数和简单转向点的常微分方程，提出不含Airy函数导数的重新表述一致渐近展开，进而得到对大阶数所有零点一致有效的贝塞尔函数及其导数零点的显式初等系数渐近展开。

On the higher-order differential equations for the generalized Laguerre polynomials and Bessel functions

In the enduring, fruitful research on spectral differential equations with polynomial eigenfunctions, Koornwinder's generalized Laguerre polynomials are playing a prominent role. Being orthogonal on the positive half-line with respect to the Laguerre weight and an additional point mass $N \ge 0$ at the origin, these polynomials satisfy, for any $α\in\mathbb{N}_{0}$, a linear differential equation of order $2α+4$. In the present paper we establish a new elementary representation of the corresponding 'Laguerre-type' differential operator and show its symmetry with respect to the underlying weigh…

An expansion for the sum of a product of an exponential and a Bessel function

We examine convergent representations for the sum of a decaying exponential and a Bessel function in the form \[\sum_{n=1}^\infty \frac{e^{-an}}{(\frac{1}{2} bn)^ν}\,J_ν(bn),\] where $J_ν(x)$ is the Bessel function of the first kind of order $ν>-1/2$ and $a$, $b$ are positive parameters. By means of a double Mellin-Barnes integral representation we obtain a convergent asymptotic expansion that enables the evaluation of this sum in the limit $a\to 0$ with $b<2π$ fixed. A similar result is derived for the sum when the Bessel function is replaced by the modified Bessel function $K_ν(x)$. The alte…

广义贝塞尔多项式在单位圆上的正正交化权函数

Positive orthogonalizing weights on the unit circle for the generalized Bessel polynomials

本文针对广义贝塞尔多项式，在特定参数条件下构造了单位圆上的正连续权函数，建立了不同于标准OPUC的正交关系，并给出其应用。

关于超球贝塞尔函数导数零点的一个注记

A note on zeros of derivatives of ultraspherical Bessel functions

本文研究超球贝塞尔函数导数正零点的渐近性质，针对任意固定参数 ν≥0、δ∈R 和 x>0，推导出当 k 趋于无穷时第 k 个正零点的渐近展开式。

Mutual Interlacing and Eulerian-like Polynomials for Weyl Groups

We use the method of mutual interlacing to prove two conjectures on the real-rootedness of Eulerian-like polynomials: Brenti's conjecture on $q$-Eulerian polynomials for Weyl groups of type $D$, and Dilks, Petersen, and Stembridge's conjecture on affine Eulerian polynomials for irreducible finite Weyl groups. For the former, we obtain a refinement of Brenti's $q$-Eulerian polynomials of type $D$, and then show that these refined Eulerian polynomials satisfy certain recurrence relation. By using the Routh--Hurwitz theory and the recurrence relation, we prove that these polynomials form a mutual…

Asymptotic expoansions of mathieu-Bessel series. I

We consider the asymptotic expansion of the Mathieu-Bessel series \[S_ν(a,b)=\sum_{n=1}^\infty \frac{n^γJ_ν(nb/a)}{(n^2+a^2)^μ}, \qquad (μ, b>0,\ γ, ν\in {\bf R})\] as $a\to+\infty$ with the other parameters held fixed, where $J_ν(x)$ is the Bessel function of the first kind of order $ν$. A special case arises when $γ+ν$ is a positive even integer, where the expansion comprises finite algebraic terms together with an exponentially small expansion. Numerical examples are presented to illustrate the accuracy of the various expansions. The expansion of the alternating variant of $S_ν(a,b)$ is con…

与Hahn-Exton q-Bessel函数有关的正交多项式和Laurent多项式

Orthogonal polynomials and Laurent polynomials related to the Hahn-Exton q-Bessel function

与Hahn-Exton $q$ -Bessel函数相关的Laurent多项式是Lommel多项式的$q$ -类似物，由Koelink和Swarttouw引入.给出了Laurent $q$ -Lommel多项式与之正交的显式强矩泛函。强矩泛函产生两个正定矩泛函。对于相应的正交多项式集的正交性措施确定使用三项递归关系作为起点。第二类切比雪夫多项式与Laurent $q$ -Lommel多项式及相关函数之间的关系被用来获得后者的估计。

An expansion for the sum of a product of an exponential and a Bessel function. II

We examine the sum of a decaying exponential (depending non-linearly on the summation index) and a Bessel function in the form \[\sum_{n=1}^\infty e^{-an^p}\frac{J_ν(an^px)}{(an^px/2)^ν}\qquad (x>0),\] in the limit $a\to0$, where $J_ν(z)$ is the Bessel function of the first kind of real order $ν$ and $a$ and $p$ are positive parameters. By means of a Mellin transform approach we obtain an asymptotic expansion that enables the evaluation of this sum in the limit $a\to 0$. A similar result is derived for the sum when the Bessel function is replaced by the modified Bessel function $I_ν(z)$ when $…

Positive intertwiners for Bessel functions of type B

Let $V_k$ denote Dunkl's intertwining operator for the root sytem $B_n$ with multiplicity $k=(k_1,k_2)$ with $k_1\geq 0, k_2>0$. It was recently shown that the positivity of the operator $V_{k^\prime\!,k} =V_{k^\prime}\circ V_k^{-1}$ which intertwines the Dunkl operators associated with $k$ and $k^\prime=(k_1+h,k_2)$ implies that $h\in[k_2(n-1),\infty[\,\cup\,(\{0,k_2,\ldots,k_2(n-1)\}-\mathbb Z_+)$. This is also a necessary condition for the existence of positive Sonine formulas between the associated Bessel functions. In this paper we present two partial converse positive results: For $k_1 \…

第一类贝塞尔函数$J_ν(x)$零点的一致渐近逼近的误差界

Error bounds for a uniform asymptotic approximation of the zeros of the Bessel function $J_ν(x)$

针对正阶$ν$的第一类贝塞尔函数$J_ν(x)$的正零点$j_{ν,m}$，研究了一种对$m$和$ν$中一个或两个无界时一致有效的近期渐近展开式，推导了其前三项与$j_{ν,m}$差值的显式上下误差界，且误差界接近首个被忽略项的值。

Radon Transform on spheres and generalized Bessel function associated with dihedral groups

Motivated by Dunkl operators theory, we consider a generating series involving a modified Bessel function and a Gegenbauer polynomial, that generalizes a known series already considered by L. Gegenbauer. We actually use inversion formulas for Fourier and Radon transforms to derive a closed formula for this series when the parameter of the Gegenbauer polynomial is a strictly positive integer. As a by-product, we get a relatively simple integral representation for the generalized Bessel function associated with even dihedral groups when both multiplicities sum to an integer. In particular, we re…

The radius of starlikeness of normalized Bessel functions of the first kind

In this note our aim is to determine the radius of starlikeness of the normalized Bessel functions of the first kind for three different kinds of normalization. The key tool in the proof of our main result is the Mittag-Leffler expansion for Bessel functions of the first kind and the fact that, according to Ismail and Muldoon [IM2], the smallest positive zeros of some Dini functions are less than the first positive zero of the Bessel function of the first kind.

关于一些涉及第一类贝塞尔函数零点的无穷级数的黎卡提型递归

Riccati type recursions for some infinite series involving zeros of Bessel functions of the first kind

研究涉及第一类贝塞尔函数正零点的无穷级数，针对现有计算方法，提出一种依赖米塔格 - 莱夫勒展开、黎卡提微分方程及泰勒级数系数的递归算法，可产生级数值，有助于处理类似特殊函数零点无穷级数问题。

根系统Bessel函数在秩趋于无穷时的极限

Limits of Bessel functions for root systems as the rank tends to infinity

本文研究类型A和类型B根系统Bessel函数在秩趋于无穷时的渐近极限，刻画了极限函数与Vershik-Kerov型谱参数序列，统一并推广了随机矩阵理论中β-系综及无穷维Hermitian矩阵空间球函数逼近的相关结果。

虚阶正自变量修正贝塞尔微分方程的解的计算

Computing solutions of the modified Bessel differential equation for imaginary orders and positive arguments

本文提出多种方法计算虚阶正自变量修正贝塞尔函数及其导数，并给出Fortran 77实现，为相关微分方程提供数值解。

An extension of the Hermite-Biehler theorem with application to polynomials with one positive root

If a real polynomial $f(x)=p(x^2)+xq(x^2)$ is Hurwitz stable (every root if $f$ lies in the open left half-plane), then the Hermite-Biehler Theorem says that the polynomials $p(-x^2)$ and $q(-x^2)$ have interlacing real roots. We extend this result to general polynomials by giving a lower bound on the number of real roots of $p(-x^2)$ and $q(-x^2)$ and showing that these real roots interlace. This bound depends on the number of roots of $f$ which lie in the left half plane. Another classical result in the theory of polynomials is Descartes' Rule of Signs, which bounds the number of positive ro…

An asymptotic for the K-Bessel function using the saddle-point method

Using the saddle-point method, we compute an asymptotic, as $y \rightarrow \infty$, for the $K$-Bessel function $K_{r + i t}(y)$ with positive, real argument $y$ and of large complex order $r+it$ where $r$ is bounded and $t = y \sin θ$ for a fixed parameter $0\leq θ\leq π/2$ or $t= y \cosh μ$ for a fixed parameter $μ>0$. Our method gives an illustrative proof, using elementary tools, of this known result and explains how these asymptotics come about. As part of our proof, we prove a new result, namely a novel integral representation for $K_{r + i t}(y)$ in the case $t= y \cosh μ$. This integra…

反广义Bessel多项式零点的一致渐近近似与数值计算

Uniform Asymptotic approximation and numerical evaluation of the Reverse Generalized Bessel Polynomial zeros

本文推导了大次数反广义Bessel多项式零点的一致渐近展开，证明其在中等或大 $n$ 值时高精度且一致有效，并以此作为高效数值算法的初始值来计算剩余复数零点。

Functional Estimates for Derivatives of the Modified Bessel Function $K_{0}$ and related Exponential Functions

Let $K_{0}$ denote the modified Bessel function of second kind and zeroth order. In this paper we will studying the function $\tildeω_{n}\left( x\right) :=\frac{\left( -x\right) ^{n}K_{0}^{\left( n\right) }\left( x\right) }{n!}$ for positive argument. The function $\tildeω_{n}$ plays an important role for the formulation of the wave equation in two spatial dimensions as a retarded potential integral equation. We will prove that the growth of the derivatives $\tildeω_{n}^{\left( m\right) }$ with respect to $n$ can be bounded by $O\left( \left( n+1\right) ^{m/2}\right) $ while for small and larg…

q-正类型函数的性质

Functions of q-positive type

本文研究了$\mathcal{L}_{q,1,v}$中q-贝塞尔傅里叶变换为正函数的子空间，并应用此结果给出了q-版本的博内尔定理。

Liouville-Green expansions of exponential form, with an application to modified Bessel functions

Linear second order differential equations of the form $d^{2}w/dz^{2}-\left \{ {u^{2}f\left( u,z\right) +g\left( z\right) }\right\} w=0$ are studied, where $\left| u\right| \rightarrow \infty $ and $z$ lies in a complex bounded or unbounded domain $\mathbf{D}$. If $f\left( u,z\right) $ and $g\left( z\right) $ are meromorphic in $\mathbf{D}$, and $f\left(u,z\right) $ has no zeros, the classical Liouville-Green/WKBJ approximation provides asymptotic expansions involving the exponential function. The coefficients in these expansions either multiply the exponential, or in an alternative form appea…

修正贝塞尔函数比值的里卡蒂简化：伯恩斯坦正性、精确证书和转移障碍

Riccati Reductions for Modified Bessel Ratios: Bernstein Positivity, Exact Certificates, and Transfer Obstructions

本文通过里卡蒂简化研究修正贝塞尔函数比值的正性、精确证书及转移障碍，反驳了相关单调性问题。

矩阵锥上的Bessel卷积

Bessel convolutions on matrix cones

本文引入了矩阵空间$M_{p,q}(\b F)$上径向有界Borel测度的卷积代数与$p\geq q$的除代数$\b F = \b R, \b C$或$\b H$之一上的正半定矩阵锥上的保概率卷积代数.在这种情况下，辐射性意味着在从左边开始的酉群$U_p(\b F)$的作用下的不变性。我们得到了一系列连续的交换超群，其特征标由矩阵辐角的Bessel函数给出。我们的结果推广了著名的结构在秩一的情况下，即Bessel-Kingman超群的正实数线，到一个更高的秩设置。在第二部分的文件中，我们研究结构只依赖于矩阵光谱。在$r\mapsto \text{spec}(r)$映射下，基础矩阵锥上的卷积在$B_q$型Weyl腔上诱导出一系列连续的超群卷积.这些字符现在是Dunkl型贝塞尔函数。这些卷积代数上的Weyl室自然延伸的调和分析与格拉斯曼流形$U(p,q)/(U_p\times U_q)$在$\b F$的Cartan运动群。

修正贝塞尔函数比值的最优精度简单界

Simple bounds with best possible accuracy for ratios of modified Bessel functions

本文刻画了修正贝塞尔函数比值的最优简单界，通过相切条件确定参数，并给出有限切点处的单参数界族。

复合贝塞尔函数 $x^\alpha J_{\nu}\left( \beta x^{\gamma}\right) $ 的对数凹性

On the log-concavity of the composite Bessel function $x^αJ_{ν}\left( βx^γ\right) $

本文研究复合贝塞尔函数 $x^\alpha J_{\nu}(\beta x^\gamma)$ 的对数凹性，通过分析 $v(F)$ 的正性给出判定准则，适用于正参数 $\beta,\gamma$ 和实参数 $\alpha,\nu$。

Recurrence relations and vector equilibrium problems arising from a model of non-intersecting squared Bessel paths

In this paper we consider the model of $n$ non-intersecting squared Bessel processes with parameter $α$, in the confluent case where all particles start, at time $t=0$, at the same positive value $x=a$, remain positive, and end, at time $T=t$, at the position $x=0$. The positions of the paths have a limiting mean density as $n\to\infty$ which is characterized by a vector equilibrium problem. We show how to obtain this equilibrium problem from different considerations involving the recurrence relations for multiple orthogonal polynomials associated with the modified Bessel functions. We also ex…

Non-intersecting squared Bessel paths with one positive starting and ending point

We consider a model of $n$ non-intersecting squared Bessel processes with one starting point $a>0$ at time t=0 and one ending point $b>0$ at time $t=T$. After proper scaling, the paths fill out a region in the $tx$-plane. Depending on the value of the product $ab$ the region may come to the hard edge at 0, or not. We formulate a vector equilibrium problem for this model, which is defined for three measures, with upper constraints on the first and third measures and an external field on the second measure. It is shown that the limiting mean distribution of the paths at time $t$ is given by the …

一种用于快速数值计算实阶次和实参数贝塞尔函数的算法

An algorithm for the rapid numerical evaluation of Bessel functions of real orders and arguments

本文提出一种快速计算实阶次和实参数贝塞尔函数的算法，通过预计算对数并利用分段双变量切比雪夫展开，提高了计算效率并避免了数值溢出问题。

反向广义贝塞尔多项式的简化艾里函数渐近展开

Simplified Airy function Asymptotic expansions for Reverse Generalised Bessel Polynomials

本文为反向广义贝塞尔多项式推导出基于艾里函数和 Liouville-Green 表示的简化均匀渐近展开，并给出系数函数的显式递推计算，适用于大阶数下包含转向点的广泛参数范围。

不相交平方Bessel路与修正Bessel权的多重正交多项式

Non-intersecting squared Bessel paths and multiple orthogonal polynomials for modified Bessel weights

我们研究了一个融合情形下的$n$非相交平方Bessel过程模型：所有路径在时间$t = 0$以相同的正值$x = a$开始，保持正值，并且条件性地在时间$t = T$以$x = 0$结束.在极限$n \to \infty$中，经过适当的重新缩放，路径填充了我们明确描述的$tx$平面中的一个区域。特别是，路径最初远离硬边在$x = 0$，但在一定的关键时刻$t^*$的最小路径击中硬边，从那时起就坚持它。对于$t \neq t^*$，我们得到通常的缩放限制随机矩阵理论，即正弦，艾里，贝塞尔内核。一个关键的事实是，在任何时候$t$的路径的位置构成一个多正交多项式系综，对应于两个修改的贝塞尔型权重的系统。因此，有一个$3 \times 3$矩阵值的黎曼-希尔伯特问题，我们分析了在大$n$极限使用Deift-Zhou最速下降法。在黎曼-希尔伯特分析中有一些新的成分是独立的利益。

Dunkl核的一个正径向积公式

A positive radial product formula for the Dunkl kernel

与根系相关的广义Bessel函数的Dunkl算子的非负重数参数具有正积公式，这是一个公开的猜想.本文证明了这一猜想的一个部分结果，即广义Bessel函数的非对称对应Dunkl核的正径积公式。因此，径向意味着乘积公式中的一个因子被其在球面上的平均值所取代。这个乘积公式的关键是Dunkl型球面平均算子的一个正性结果。它也可以解释为径向函数的Dunkl型广义平移是保正的。作为应用，我们构造了与径向概率分布相关的Dunkl型齐次马氏过程。

On a General Method for Resolving Integrals of Multiple Spherical Bessel Functions Against Power Laws into Distributions

We here present a method of performing integrals of products of spherical Bessel functions (SBFs) weighted by a power-law. Our method, which begins with double-SBF integrals, exploits a differential operator $\hat{D}$ defined via Bessel's differential equation. Application of this operator raises the power-law in steps of two. We also here display a suitable base integral expression to which this operator can be applied for both even and odd cases. We test our method by showing that it reproduces previously-known solutions. Importantly, it also goes beyond them, offering solutions in terms of …

Asymptotic relation for zeros of cross-product of Bessel functions and applications

Let $a_{ν,k}$ be the $k$-th positive zero of the cross-product of Bessel functions $J_ν(R z) Y_ν(z) - J_ν(z) Y_ν(R z)$, where $ν\geq 0$ and $R>1$. We derive an initial value problem for a first order differential equation whose solution $α(x)$ characterizes the limit behavior of $a_{ν,k}$ in the following sense: $$ \lim_{k \to \infty} \frac{a_{kx,k}}{k} = α(x), \quad x \geq 0. $$ Moreover, we show that $$ a_{ν,k} < \frac{πk}{R-1} + \frac{πν}{2R}. $$ We use $α(x)$ to obtain an explicit expression of the Pleijel constant for planar annuli and compute some of its values.

Eisenstein series and an asymptotic for the $K$-Bessel function

We produce an estimate for the $K$-Bessel function $K_{r + i t}(y)$ with positive, real argument $y$ and of large complex order $r+it$ where $r$ is bounded and $t = y \sin θ$ for a fixed parameter $0\leq θ\leq π/2$ or $t= y \cosh μ$ for a fixed parameter $μ>0$. In particular, we compute the dominant term of the asymptotic expansion of $K_{r + i t}(y)$ as $y \rightarrow \infty$. When $t$ and $y$ are close (or equal), we also give a uniform estimate. As an application of these estimates, we give bounds on the weight-zero (real-analytic) Eisenstein series $E_0^{(j)}(z, r+it)$ for each inequivalen…

Olshanski spherical functions for infinite dimensional motion groups of fixed rank

Consider the Gelfand pairs $(G_p,K_p):=(M_{p,q} \rtimes U_p,U_p)$ associated with motion groups over the fields $\mathbb F=\mathbb R,\mathbb C,\mathbb H$ with $p\geq q$ and fixed $q$ as well as the inductive limit $p\to\infty$,the Olshanski spherical pair $(G_\infty,K_\infty)$. We classify all Olshanski spherical functions of $(G_\infty,K_\infty)$ as functions on the cone $Π_q$ of positive semidefinite $q\times q$-matrices and show that they appear as (locally) uniform limits of spherical functions of $(G_p,K_p)$ as $p\to\infty$. The latter are given by Bessel functions on $Π_q$. Moreover, we …