K-Bessel Functions Research Articles

Hankel transform, [formula omitted]-Bessel functions and zeta distributions in the Dunkl setting

We study analytic properties of a Hankel transform for the type A Dunkl setting with arbitrary multiplicity parameter k≥0 which goes back to Baker and Forrester and, in an earlier symmetrized version, to Macdonald. Moreover, we introduce a Dunkl analogue of the Bessel functions and K-Bessel functions, generalizing those of a symmetric cone which occur when the multiplicity parameter k is chosen to be twice the Peirce dimension constant of the cone. In particular, we show that our K-Bessel function is a solution of a type B system of Dunkl operators which is a generalization of the Bessel system on a symmetric cone. Furthermore, we define zeta distributions and prove a functional equation relating them with their type B Dunkl transform. In the case of symmetric cones the aforementioned functional equation reduces to the known functional equation between zeta distributions and their Fourier transform. For the proof of this functional equation, the K-Bessel function and its analytic properties turn out to be crucial. Finally, we examine which of the zeta distributions are given by a positive measure.

Radii of starlikeness and convexity of generalized k-Bessel functions

Abstract The main purpose of this study is to determine the radii of starlikeness and convexity of the generalized k-Bessel functions for three different kinds of normalization in such a way that the resulting functions are analytic in the unit disk of the complex plane. The characterization of entire functions from Laguerre–Pólya class plays a significant role in this paper. Moreover, the interlacing properties of the zeros of the k-Bessel function and its derivative is also useful in the proof of the main results. By making use of the Euler–Rayleigh inequalities for the real zeros of the generalized k-Bessel function, we obtain some tight lower and upper bounds for the radii of starlikeness and convexity of order zero.

Interacting diffusions on positive definite matrices

We consider systems of Brownian particles in the space of positive definite matrices, which evolve independently apart from some simple interactions. We give examples of such processes which have an integrable structure. These are related to K-Bessel functions of matrix argument and multivariate generalisations of these functions. The latter are eigenfunctions of a particular quantisation of the non-Abelian Toda lattice.

Comments on Deuring's zero-spacing phenomenon

We codify some aspects of Deuring's zero-spacing phenomenon, namely that in the presence of an exceptional Landau/Siegel zero, various product L-functions have their low-height zeros lie in a very regular pattern, nearly an arithmetic progression on the central line. This piece of “analytic number theory folklore” seems to have had its actual proof escape the literature in generality, yet we still claim to provide a “different” proof in that we use Lavrik's general method for approximate functional equations, rather than (say, in the imaginary quadratic case) a decomposition into Epstein ζ-functions and analysis with K-Bessel functions and Kloosterman sums.

Differential equation and inequalities of the generalized k-Bessel functions

In this paper, we introduce and study a generalization of the k-Bessel function of order ν given by \t\t\tWν,ck(x):=∑r=0∞(−c)rΓk(rk+ν+k)r!(x2)2r+νk.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\mathtt{W}^{\\mathtt{k}}_{\\nu , c}(x):= \\sum_{r=0}^{\\infty } \\frac{(-c)^{r}}{\\Gamma_{\\mathtt{k}} ( r \\mathtt{k} +\\nu +\\mathtt{k} ) r!} \\biggl( \\frac{x}{2} \\biggr) ^{2r+\\frac{\\nu }{\\mathtt{k}}}. $$\\end{document} We also indicate some representation formulae for the function introduced. Further, we show that the function mathtt{W}^{ mathtt{k}}_{nu , c} is a solution of a second-order differential equation. We investigate monotonicity and log-convexity properties of the generalized k-Bessel function mathtt{W}^{mathtt{k}} _{nu , c}, particularly, in the case c=-1. We establish several inequalities, including a Turán-type inequality. We propose an open problem regarding the pattern of the zeroes of mathtt{W}^{ mathtt{k}}_{nu , c}.

Bessel operators on Jordan pairs and small representations of semisimple Lie groups

We provide a uniform construction of L2-models for all small unitary representations in degenerate principal series of semisimple Lie groups which are induced from maximal parabolic subgroups with abelian nilradical. This generalizes previous constructions to the case of a maximal parabolic subgroup which is not necessarily conjugate to its opposite, and hence the previously used Jordan algebra methods have to be generalized to Jordan pairs. The crucial ingredients for the construction of the L2-models are the Lie algebra action and the spherical vector. Working in the so-called Fourier transformed picture of the degenerate principal series, the Lie algebra action is given in terms of Bessel operators on Jordan pairs. We prove that precisely for those parameters of the principal series where small quotients occur, the Bessel operators are tangential to certain submanifolds. Further, we show that the small quotients are unitarizable if and only if these submanifolds carry equivariant measures. In this case we can express the spherical vectors in terms of multivariable K-Bessel functions, and some delicate estimates for these Bessel functions imply the existence of the L2-models.

$SL(2, \mathbb{Z})$-invariance and D-instanton contributions to the $D^6 R^4$ interaction

The modular invariant coefficient of theD 6 R 4 interaction in the low energy expansion of type IIB string theory has been conjec- tured to be a solution of an inhomogeneous Laplace eigenvalue equa- tion, obtained by considering the toroidal compactification of two-loop Feynman diagrams of eleven-dimensional supergravity. In this paper we determine the exact SL(2,Z)-invariant solution f(x + iy) to this differ- ential equation satisfying an appropriate moderate growth condition as y ! 1 (the weak coupling limit). The solution is presented as a Fourier series with modes b fn(y)e 2�inx , where the mode coefficients,b fn(y) are bi- linear in K-Bessel functions. Invariance under SL(2,Z) requires these modes to satisfy the nontrivial boundary condition b fn(y) = O(y −2 ) for small y, which uniquely determines the solution. The large-y expansion of f(x + iy) contains the known perturbative (power-behaved) terms, together with precisely-determined exponentially decreasing contribu- tions that have the form expected of D-instantons, anti-D-instantons and D-instanton/anti-D-instanton pairs.

The chiral Gaussian two-matrix ensemble of real asymmetric matrices

We solve a family of Gaussian two-matrix models with rectangular N × (N + ν) matrices, having real asymmetric matrix elements and depending on a non-Hermiticity parameter μ. Our model can be thought of as the chiral extension of the real Ginibre ensemble, relevant for Dirac operators in the same symmetry class. It has the property that its eigenvalues are either real, purely imaginary or come in complex conjugate eigenvalue pairs. The eigenvalue joint probability distribution for our model is explicitly computed, leading to a non-Gaussian distribution including K-Bessel functions. All n-point density correlation functions are expressed for finite N in terms of a Pfaffian form. This contains a kernel involving Laguerre polynomials in the complex plane as a building block which was previously computed by the authors. This kernel can be expressed in terms of the kernel for complex non-Hermitian matrices, generalizing the known relation among ensembles of Hermitian random matrices. Compact expressions are given for the density at finite N as an example, as well as its microscopic large-N limits at the origin for fixed ν at strong and weak non-Hermiticity.

On Bessel series expressions for some lattice sums: II

In part I (Kanemitsu S et al 2003 J. Northwest University) we have made explicit use of the Mellin–Barnes integrals to prove the Chowla–Selberg-type Bessel series expressions for zeta-functions associated with lattice structures. In this paper we shall make implicit use of Mellin–Barnes integrals, as embedded in our theory of modular relations and functional equations, to reveal relationships between the structure of Madelung constants of the NaCl and CsCl lattices. Namely, we shall elucidate the relation between the structures of the NaCl lattice and those of the CsCl lattice, so to speak using the symmetry of the zeta-function, i.e. using their functional equations. Thus we shall emphasize the symmetry properties of the zeta-functions, restoring the Schlomilch series and Hardy's theory of K-Bessel functions, to prove the functional equations, and then to prove the recurrence relations for the lattice zeta-functions.

The Chowla-Selberg Method for Fourier Expansion of Higher Rank Eisenstein Series

AbstractThe terms of maximal rank in Fourier expansions of Eisenstein series for GL(n, ℤ) are obtained by an analogue of a method of Chowla and Selberg. The coefficients involve matrix analogues of divisor functions as well as K-Bessel functions for GL(n). The discussion involves a few properties of Hecke operators.

Special Functions for the Symmetric Space of Positive Matrices

Section 1 of the paper gives three applications of a basic principle that eigenfunctions of invariant differential operators are eigenfunetions of invariant integral operators. The first application is a derivation of a noneuclidean analogue of the Poisson summation formula, the second is the evaluation of an integral of Muirhead which has been of interest in multivariate statistics, the third is the evaluation of gammaa type integrals arising in the theory of Eisenstein series for the general linear group. Section 2 of the paper concerns K-Bessel functions for the general linear group. It relates such functions with gamma functions for the symplectic group and applies the theory to the estimation of Fourier coefficients of automorphic forms for ${\operatorname{GL}}(3,\mathbb{Z})$.

Noneuclidean Harmonic Analysis

Analysis on Lie groups and their homogeneous spaces has seen a recent flowering, thanks to the work of Harish-Chandra, Helgason, Selberg and many others. The purpose of this paper is to give an elementary discussion of Fourier analysis on the noneuclidean upper half plane H; that is, the spectral resolution of the noneuclidean Laplace operator $\Delta $. This leads to a study of the elementary eigenfunctions of $\Delta $. Such eigenfunctions can be viewed as analogues of the trigonometric functions in euclidean Fourier analysis. The elementary eigenfunctions of $\Delta $ on H are Legendre (conical or spherical) functions, K-Bessel functions and Eisenstein series. The invariance of $\Delta $ under the special linear group $G = SL(2,\mathbb{R})$, of $2 \times 2$ real matrices of determinant l, gives a way of recognizing the spectral measure in the noneuclidean Fourier inversion formula, once one knows the asymptotics and functional equations of the elementary eigenfunctions of $\Delta $. In fact, this method allows a unified treatment of the inversion formulas for the standard integral transforms of mathematical physics. The connection between Fourier analysis on H and that on the fundamental domain $\Gamma \backslash H$, $\Gamma = $ a discrete subgroup of G (e.g., $\Gamma = $ the modular group $SL(2,\mathbb{Z})$ of $2 \times 2$ integer matrices of determinant 1) provides an analogue of the Poisson summation formula. Applications to the solution of the heat equation on H and $\Gamma \backslash H$ are considered, as well as the distribution of eigenvalues of the noneuclidean Laplacian on $L^2 (\Gamma \backslash H)$.