Semyon Yakubovich
Discrete analogs of the index transforms, involving Bessel and Lommel functions are introduced and investigated. The corresponding inversion theorems for suitable classes of functions and sequences are established.
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For the Bessel function \begin{equation} \label{bessel} J_ν(z) = \sum\limits_{k=0}^{\infty} \frac{(-1)^k \left( \frac{z}{2} \right)^{ν+2k}}{k! Γ(ν+1+k)} \end{equation} there exist several $q$-analogues. The oldest $q$-analogues of the Bessel function were introduced by F. H. Jackson at the beginning of this century, see M. E. H. Ismail \cite{Is1} for the appropriate references. Another $q$-analogue of the Bessel function has been introduced by W. Hahn in a special case and by H. Exton in full generality, see R. F. Swarttouw \cite{Sw1} for a historic overview. Here we concentrate on properties …
We show that the Bessel function asymptotic relation $J_ν^2(z) + J_{ν+ 1}^2 (z) \sim 2/(πz)$ of Lommel is valid when $z$ is real but can fail otherwise.
Laurent polynomials related to the Hahn-Exton $q$-Bessel function, which are $q$-analogues of the Lommel polynomials, have been introduced by Koelink and Swarttouw. The explicit strong moment functional with respect to which the Laurent $q$-Lommel polynomials are orthogonal is given. The strong moment functional gives rise to two positive definite moment functionals. For the corresponding sets of orthogonal polynomials the orthogonality measure is determined using the three-term recurrence relation as a starting point. The relation between Chebyshev polynomials of the second kind and the Laure…
We derive new approximate representations of the Lommel functions in terms of the Scorer function and approximate representations of the first derivative of the Lommel functions in terms of the derivative of the Scorer function. Using the same method we obtain previously known approximate representations of the Nicholson type for Bessel functions and their first derivatives. We study also for what values of the parameters our representations have reasonable accuracy.
Using a differential equation approach asymptotic expansions are rigorously obtained for Lommel, Weber, Anger-Weber and Struve functions, as well as Neumann polynomials, each of which is a solution of an inhomogeneous Bessel equation. The approximations involve Airy and Scorer functions, and are uniformly valid for large real order $ν$ and unbounded complex argument $z$. An interesting complication is the identification of the Lommel functions with the new asymptotic solutions, and in order to do so it is necessary to consider certain sectors of the complex plane, as well as introduce new form…
In 1993 Delest and Fédou showed that a generating function for connected skew shapes is given as a ratio $J_{ν+1}/J_ν$ of the Hahn--Exton $q$-Bessel functions when a parameter $ν$ is zero. They conjectured that when $ν$ is a nonnegative integer the coefficients of the generating function are rational functions whose numerator and denominator are polynomials in $q$ with nonnegative integer coefficients, which is a $q$-analog of Kishore's 1963 result on Bessel functions. The first main result of this paper is a proof of the conjecture of Delest and Fédou. The second main result is a refinement o…
The radius of convexity of two normalized Bessel functions of the first kind are determined in the case when the order is between $-2$ and $-1.$ Our methods include the minimum principle for harmonic functions, the Hadamard factorization of some Dini functions, properties of the zeros of Dini functions via Lommel polynomials and some inequalities for complex and real numbers.
In this paper we consider some normalized Bessel, Struve and Lommel functions of the first kind, and by using the Euler-Rayleigh inequalities for the first positive zeros of combination of special functions we obtain tight lower and upper bounds for the radii of starlikeness of these functions. By considering two different normalization of Bessel and Struve functions we give some inequalities for the radii of convexity of the same functions. On the other hand, we show that the radii of univalence of some normalized Struve and Lommel functions are exactly the radii of starlikeness of the same f…
Tight lower and upper bounds for the radius of univalence of some normalized Bessel, Struve and Lommel functions of the first kind are obtained via Euler-Rayleigh inequalities. It is shown also that the radius of univalence of the Struve functions is greater than the corresponding radius of univalence of Bessel functions. Moreover, by using the idea of Kreyszig and Todd, and Wilf it is proved that the radii of univalence of some normalized Struve and Lommel functions are exactly the radii of starlikeness of the same functions. The Laguerre-Pólya class of entire functions plays an important rol…
In this paper our aim is to deduce some sufficient (and necessary) conditions for the close-to-convexity of some special functions and their derivatives, like Bessel functions, Struve functions, and a particular case of Lommel functions of the first kind, which can be expressed in terms of the hypergeometric function ${}_1F_2$. The key tool in our proofs is a result of Shah and Trimble about transcendental entire functions with univalent derivatives. Moreover, a known result of Pólya on entire functions, the infinite product representations and some results on zeros of Bessel, Struve and Lomme…
The modified Lommel function $t_{μ,ν}(x)$ is an important special function, but to date there has been little progress on the problem of obtaining functional inequalities for $t_{μ,ν}(x)$. In this paper, we advance the literature substantially by obtaining a simple two-sided inequality for the ratio $t_{μ,ν}(x)/t_{μ-1,ν-1}(x)$ in terms of the ratio $I_ν(x)/I_{ν-1}(x)$ of modified Bessel functions of the first kind, thereby allowing one to exploit the extensive literature on bounds for this ratio. We apply this result to obtain two-sided inequalities for the condition numbers $xt_{μ,ν}'(x)/t_{μ…
On the generalized interlacing property for the zeros of Bessel functions
本文研究了 Bessel 函数在 $|ν-μ|>2$ 时的广义交错性质及其与 Lommel 多项式零点的补偿作用，并将该性质扩展至柱函数与 Bessel 函数导数，同时证明了 Siegel 对 Bourget 假设的扩展无法推广至任意实数。
Three-Term Recurrence Relations for Confluent Basic Hypergeometric Series with Applications to q-Bessel Functions
本文推导了连通基本超几何级数的三项递推关系，并应用于q-贝塞尔函数的递推关系研究。
This paper offers a new and complete description of subnormal solutions of certain non-homogeneous second order periodic linear differential equations first studied by Gundersen and Steinbart in 1994. We have established a previously unknown relation that the general solutions (\textit{i.e.}, whether subnormal or not) of the DEs can be solved explicitly in terms of classical special functions, namely the Bessel, Lommel and Struve functions, which are important because of their numerous physical applications. In particular, we show that the subnormal solutions are written explicitly in terms of…
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。
A class of orthogonal polynomials associated with Coulomb wave functions is introduced. These polynomials play a role analogous to that the Lommel polynomials do in the theory of Bessel functions. The measure of orthogonality for this new class is described explicitly. In addition, the orthogonality measure problem is also discussed on a more general level. Apart of this, various identities derived for the new orthogonal polynomials may be viewed as generalizations of some formulas known from the theory of Bessel functions. A key role in these derivations is played by a Jacobi (tridiagonal) ma…
Discrete analogs of the index transforms, involving Bessel and the modified Bessel functions are introduced and investigated. The corresponding inversion theorems for suitable classes of functions and sequences are established.
Discrete analogs of the index transforms with squares of Bessel functions of the first and second kind $J_ν(z),\ Y_ν(z)$ are introduced and investigated. The corresponding inversion theorems for suitable classes of functions and sequences are established.
New index transforms are investigated, which contain as the kernel products of the Bessel and modified Bessel functions. Mapping properties and invertibility in Lebesgue spaces are studied for these operators. Relationships with the Kontorovich-Lebedev and Fourier cosine transforms are established. Inversion theorems are proved. As an application, a solution of the initial value problem for the fourth order partial differential equation, involving the Laplacian is presented.
New index transforms, involving the square of Bessel functions of the first kind as the kernel are considered. Mapping properties such as the boundedness and invertibility are investigated for these operators in the Lebesgue spaces. Inversion theorems are proved. As an interesting application, a solution to the initial value problem for the third order partial differential equation, involving the Laplacian, is obtained.
New index transforms of the Lebedev type are investigated. It involves the real part of the product of the modified Bessel functions as the kernel. The boundedness and invertibility are examined for these operators in the Lebesgue weighted spaces. Inversion theorems are proved. Important particular cases are exhibited. The results are applied to solve an initial value problem for the fourth order PDE, involving the Laplacian. Finally, it is shown that the same PDE has another fundamental solution, which is associated with the generalized Lebedev index transform, involving the square of the mod…
Discrete analogs of the Lebedev transforms with the product of the modified Bessel functions are introduced and investigated. Several expansions of suitable functions and sequences in terms of the series and integrals, involving the modified and incomplete Bessel functions are established.
This paper presents the equality of finite index sums of Bessel func- tions containing arbitrary numbers of terms. These reduce to the familiar three term recursion formulas in simple cases.
New index transforms, involving the real part of the modified Bessel function of the first kind as the kernel are considered. Mapping properties such as the boundedness and invertibility are investigated for these operators in the Lebesgue spaces. Inversion theorems are proved. As an interesting application, a solution of the initial value problem for the second order partial differential equation, involving the Laplacian, is obtained. It is noted, that the corresponding operators with the imaginary part of the modified Bessel function of the first kind lead to the familiar Kontorovich- Lebede…
In the paper we study applications of integral transforms composition method (ITCM) for obtaining transmutations via integral transforms. It is possible to derive wide range of transmutation operators by this method. Classical integral transforms are involved in the integral transforms composition method (ITCM) as basic blocks, among them are Fourier, sine and cosine-Fourier, Hankel, Mellin, Laplace and some generalized transforms. The ITCM and transmutations obtaining by it are applied to deriving connection formulas for solutions of singular differential equations and more simple non-singula…
We review some aspects of the theory of spherical Bessel functions and Struve functions by means of an operational procedure essentially of umbral nature, capable of providing the straightforward evaluation of their definite integrals and of successive derivatives. The method we propose allows indeed the formal reduction of these family of functions to elementary ones of Gaussian type. We study the problem in general terms and present a formalism capable of providing a unifying point of view including Anger and Weber functions too. The link to the multi-index Bessel functions is also briefly d…
A new index transform with the square of Whittaker's function
本文提出并研究了一种基于Whittaker函数平方的指标变换，建立了反演公式，其特例涵盖Lebedev型变换。
New biorthogonal sequences generated by index integrals of the weight functions
本文通过修正贝塞尔函数和伽马函数的索引积分生成新正交序列，并研究了广义威尔逊多项式性质。
New index transforms with Weber type kernels, consisting of products of Bessel functions of the first and second kind are investigated. Mapping properties and inversion formulas are established for these transforms in Lebesgue spaces. The results are applied to solve a boundary value problem on the wedge for a fourth order partial differential equation.
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 $…
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 …
We examine the sum of modified Bessel functions with argument depending non-linearly on the summation index given by \[S_{ν,p}(a)=\sum_{n\geq 1} (an^p/2)^{-ν} K_ν(an^p)\qquad (a>0,\ 0\leqν<1)\] as the parameter $a\to 0+$, where $p$ denotes an integer satisfying $p\geq 2$. This extends previous work for the cases $p=1$ (linear) and $p=2$ (quadratic). The expansion as $a\to0+$ consists of an infinite number of asymptotic sums involving the Riemann zeta function, which when optimally truncated lead to remainder terms that are exponentially small in the parameter $a$. The number of these exponenti…
Generalized product formulas for Whittaker's functions and a novel class of index transforms
本文建立了涉及不同指标Whittaker函数乘积的一般化乘积公式和指标变换，并给出反演公式，涵盖修正Bessel函数等特殊情形，利用复指标Kontorovich-Lebedev和Olevskii变换。
A note on the tetrahedral index and the Hahn-Exton q-Bessel function
本文旨在阐明四面体指标I_Δ(m,e)与Hahn-Exton q-Bessel函数的联系，并介绍物理数学应用于几何拓扑产生的新技术与猜想。
In this paper, we consider the normalized Bessel function of index $α> -\frac{1}{2}$, we find an integral representation of the term $x^nj_{α+n}(x)j_α(y)$. This allows us to establish a product formula for the generalized Hankel function $B^{κ,n}_λ$ on $\mathbb{R}$. $B^{κ,n}_λ$ is the kernel of the integral transform $\mathcal{F}_{κ,n}$ arising from the Dunkl theory. Indeed we show that $B^{κ,n}_λ(x)B^{κ,n}_λ(y)$ can be expressed as an integral in terms of $B^{κ,n}_λ(z)$ with explicit kernel invoking Gegenbauer polynomials for all $n\in\mathbb{N}^\ast$. The obtained result generalizes the prod…
There exist several multivariate extensions of the classical Sonine integral representation for Bessel functions of some index $μ+ ν$ with respect to such functions of lower index $μ.$ For Bessel functions on matrix cones, Sonine formulas involve beta densities $β_{μ,ν}$ on the cone and trace already back to Herz. The Sonine representations known so far on symmetric cones are restricted to continuous ranges $\Reμ, \Re ν> μ_0$, where the involved Beta densities are probability measures and the limiting index $μ_0\geq 0$ depends on the rank of the cone. It is zero only in the one-dimensional cas…
We are concerned with the zeros of the Macdonald functions or the modified Bessel functions of the second kind with real index. By using the explicit expressions for the algebraic equations satisfied by the zeros, we describe the behavior of the zeros when the index moves. Results by numerical computations are also presented.
Index theory for singular Lagrangian systems and Bessel-type differential operators
本文针对奇异拉格朗日系统发展指标理论，建立奇异Sturm Liouville算子的谱流公式与Morse指标定理，构建Bessel型微分算子的Morse指标理论并应用于引力n体问题，同时为Rellich观察到的变定义域谱现象提供新解释。
A different application of the familiar integral representation for the modifed Bessel function drives to a new Kontorovich-Lebedev-like integral transformation of a general complex index. Mapping and operational properties, a convolution operator and inversion formula are established. Solvability conditions and explicit solutions of the corresponding class of convolution integral equations are exhibited.
Schatten--Lorentz characterization of Riesz transform commutator associated with Bessel operators
本文用鞅和二进分析刻画了与Bessel算子相关的Riesz变换换位子的Schatten--Lorentz范数，建立了其与符号振荡空间及Besov范数的等价关系，并确定了临界指标为Bessel测度的下维数。
We consider mixed type multiple orthogonal polynomials associated with a system of weight functions consisting of two vectors. One vector is defined in terms of scaled modified Bessel function of the first kind $I_μ$ and $I_{μ+1}$, the other vector is defined in terms of scaled modified Bessel function of the second kind $K_ν$ and $K_{ν+1}$. We show that the corresponding mixed type multiple orthogonal polynomials exist. For the special case that each multi-index is on or close to the diagonal, basic properties of the polynomials and their linear forms are investigated, which include explicit …
Neumann series of Bessel functions in direct and inverse spherically symmetric transmission eigenvalue problems
本文提出基于贝塞尔函数诺伊曼级数（NSBF）的方法，通过刘维尔变换将球对称透射特征值问题转化为施图姆-刘维尔方程，实现实、复特征值的高精度求解，并利用两步法从谱数据中恢复区间长度和折射率，无需先验假设。
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