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

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 2023年第1期526卷

作者： Segura, Javier Univ Cantabria Dept Matemat Estadist & Comp Santander 39005 Spain

The best bounds of the form B(alpha, beta, gamma, x) = (alpha + beta 2 + gamma 2x2)/x for ratios of modified bessel functions are characterized: if alpha, beta and gamma are chosen in such a way that B(alpha, beta, gamma, x) is a sharp approximation for (13,,(x) = I,,-1(x)/I,,(x) as x-+0+ (respectively x-+ +infinity) and the graphs of the functions B(alpha, beta, gamma, x) and (13,,(x) are tangent at some x = x* > 0, then B(alpha, beta, gamma, x) is an upper (respectively lower) bound for (13,,(x) for any positive x, and it is the best possible at x*. The same is true for the ratio (13,,(x) = K,,+1(x)/K,,(x) but interchanging lower and upper bounds (and with a slightly more restricted range for nu). Bounds with maximal accuracy at 0+ and +infinity are recovered in the limits x*-+ 0+ and x*-+ +infinity, and for these cases the coefficients have simple expressions. For the case of finite and positive x* we provide uniparametric families of bounds which are close to the optimal bounds and retain their confluence properties.(c) 2023 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http:// creativecommons .org /licenses /by -nc -nd /4 .0/).

关键词： modified bessel functions Ratios Best bounds

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Convexity and concavity of the modified bessel functions of the first kind with respect to Holder means

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REVISTA DE LA REAL ACADEMIA DE CIENCIAS EXACTAS FISICAS Y NATURALES SERIE A-MATEMATICAS 2020年第2期114卷 1-14页

作者： Zhao, Tie-Hong Shi, Lei Chu, Yu-Ming Hangzhou Normal Univ Dept Math Hangzhou 311121 Peoples R China Huzhou Teachers Coll Dept Math Huzhou 313000 Peoples R China Changsha Univ Sci & Technol Sch Math & Stat Changsha 410114 Peoples R China

In this paper, we establish a necessary and sufficient condition for the convexity and concavity of the modified bessel functions of the first kind with respect to Holder means.

关键词： Convexity Concavity modified bessel functions Holder means

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Recursive Computation of Logarithmic Derivatives, Ratios, and Products of Spheroidal Harmonics and modified bessel functions and Applications

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JOURNAL OF SCIENTIFIC COMPUTING 2018年第1期75卷 128-156页

作者： Xue, Changfeng Deng, Shaozhong Yancheng Inst Technol Sch Math & Phys Yancheng 224051 Jiangsu Peoples R China Univ North Carolina Charlotte Dept Math & Stat Charlotte NC 28223 USA

Spheroidal harmonics and modified bessel functions have wide applications in scientific and engineering computing. Recursive methods are developed to compute the logarithmic derivatives, ratios, and products of the prolate spheroidal harmonics (, , , ), the oblate spheroidal harmonics (, , , ), and the modified bessel functions (, , , ) in order to avoid direct evaluation of these functions that may easily cause overflow/underflow for high degree/order and for extreme argument. Stability analysis shows the proposed recursive methods are stable for realistic degree/order and argument values. Physical examples in electrostatics are given to validate the recursive methods.

关键词： Prolate spheroidal harmonics Oblate spheroidal harmonics modified bessel functions Associated Legendre functions Continued fraction

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ON A NEUMANN-TYPE SERIES FOR modified bessel functions OF THE FIRST KIND

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PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY 2018年第5期146卷 2149-2161页

作者： Deleaval, L. Demni, N. Univ Paris Est Marne La Vallee Lab Anal & Math Appl F-77454 Marne La Vallee 2 France Univ Rennes 1 IRMAR Campus Beaulieu F-35042 Rennes France

In this paper, we are interested in a Neumann-type series for modified bessel functions of the first kind which arises in the study of Dunkl operators associated with dihedral groups and as an instance of the Laguerre semigroup constructed by Ben Said-Kobayashi-Orsted. We first revisit the particular case corresponding to the group of square-preserving symmetries for which we give two new and different proofs other than the existing ones. The first proof uses the expansion of powers in a Neumann series of bessel functions, while the second one is based on a quadratic transformation for the Gauss hypergeometric function and opens the way to derive further expressions when the orders of the underlying dihedral groups are powers of two. More generally, we give another proof of De Bie et al.'s formula expressing this series as a Phi(2)-Horn confluent hypergeometric function. In the course of the proof, we shed light on the occurrence of multiple angles in their formula through elementary symmetric functions and get a new representation of Gegenbauer polynomials.

关键词： modified bessel functions Gegenbauer polynomials generalized bessel function dihedral groups elementary symmetric functions

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Mixed type multiple orthogonal polynomials associated with the modified bessel functions and products of two coupled random matrices

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JOURNAL OF APPROXIMATION THEORY 2017年 213卷 92-119页

作者： Zhang, Lun Fudan Univ Sch Math Sci Shanghai 200433 Peoples R China Fudan Univ Shanghai Key Lab Contemporary Appl Math Shanghai 200433 Peoples R China

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-mu and I mu+1, the other vector is defined in terms of scaled modified bessel function of the second kind K-nu and K nu+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 formulas, integral representations, differential properties, limiting forms and recurrence relations. It comes out that, for specified parameters, the linear forms of these mixed type multiple orthogonal polynomials can be interpreted as biorthogonal functions encountering in recent studies of products of two coupled random matrices. This particularly implies a Riemann-Hilbert characterization of the correlation kernel, which provides an alternative way for further asymptotic analysis. (C) 2016 Elsevier Inc. All rights reserved.

关键词： Mixed type multiple orthogonal polynomials modified bessel functions Integral representations Limiting forms Recurrence relations Random matrices

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Bounds for the product of modified bessel functions

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AEQUATIONES MATHEMATICAE 2016年第4期90卷 859-870页

作者： Baricz, Arpad Masirevic, Dragana Jankov Ponnusamy, Saminathan Singh, Sanjeev Obuda Univ Inst Appl Math H-1034 Budapest Hungary Babes Bolyai Univ Dept Econ Cluj Napoca 400591 Romania Univ Osijek Dept Math Osijek 31000 Croatia MGR Knowledge City Chennai Ctr Soc Elect Transact & Secur Indian Stat Inst CIT Campus Chennai 600113 Tamil Nadu India Indian Inst Technol Madras Dept Math Chennai 600036 Tamil Nadu India

In this note our aim is to present some monotonicity properties of the product of modified bessel functions of the first and second kind. Certain bounds for the product of modified bessel functions of the first and second kind are also obtained. These bounds improve and extend known bounds for the product of modified bessel functions of the first and second kind of order zero. A new Turan type inequality is also given for the product of modified bessel functions, and some open problems are stated, which may be of interest for further research.

关键词： modified bessel functions Turan type inequalities monotonicity properties bounds

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A new type of sharp bounds for ratios of modified bessel functions

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JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 2016年第2期443卷 1232-1246页

作者： Ruiz-Antolin, Diego Segura, Javier Univ Cantabria Dept Matemat Estadist & Computac E-39005 Santander Spain

The bounds for the ratios of first and second kind modified bessel functions of consecutive orders are important quantities appearing in a large number of scientific applications. We obtain new bounds which are accurate in a large region of parameters and which are sharper than previous bounds. The new bounds are obtained by a qualitative analysis of the Riccati equation satisfied by these ratios. A procedure is considered in which the bounds obtained from the analysis of the Riccati equation are used to define a new function satisfying a new Riccati equation which yields sharper bounds. Similar ideas can be applied to other functions. (C) 2016 Elsevier Inc. All rights reserved.

关键词： modified bessel functions Bounds

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Redheffer type inequalities for modified bessel functions

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Arab Journal of Mathematical Sciences 2016年第1期22卷 38-42页

作者： Mehrez, Khaled Département de Mathématiques Istitut Préparatoire aux Etudes d'ingénieurs de Monastir Monastir 5000 Tunisia

In this short note, we give new proofs of Redheffer's inequality for modified bessel functions of first kind published by Ling Zhu (2011). In addition, using the Grosswald formula we prove new Redheffer type inequ... 详细信息

关键词： modified bessel functions Sharpening Redheffer type inequalities

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Lebedev's Type Index Transforms with the modified bessel functions

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Communications in Mathematical Analysis 2016年第2期19卷 68-81页

作者： Semyon Yakubovich

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. Boundedness properties 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 modulus of Macdonald's function, recently considered by the author.

关键词： 33C10 44A05 44A15 Fourier transform Index Transform Initial value problem Lebedev transform Mellin transform modified bessel functions

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On determinants of modified bessel functions and entire solutions of double confluent Heun equations

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NONLINEARITY 2016年第12期29卷 3857-3870页

作者： Buchstaber, V. M. Glutsyuk, A. A. Steklov Math Inst 8 Gubkina St Moscow 119991 Russia CNRS F-75700 Paris France UMPA ENS Lyon UMR 5669 Lyon France Lab JV Poncelet UMI 2615 Lyon France Natl Res Univ Higher Sch Econ Moscow Russia All Russian Sci Res Inst Phys & Radio Tech Measur Moscow Russia

We investigate the question on existence of entire solutions of well-known linear differential equations that are linearizations of nonlinear equations modeling the Josephson effect in superconductivity. We consider the modified bessel functions I-j(x) of the first kind, which are Laurent series coefficients of the analytic function family e(x/2(z+1/z)). For every l >= 1 we study the family parametrized by k,n is an element of Z(l), k(1) > .... > k(l), n(1) > ... >n(l) of (l X l)-matrix functions formed by the modified bessel functions of the first kind a(ij)(x) = Ikj-ni(x), i, j = 1,...,l. We show that their determinants f(k,n)(x) are positive for every l (3) 1, k,n is an element of Z(l) as above and x > 0. The above determinants are closely related to a sequence (indexed by l) of families of double confluent Heun equations, which are linear second order differential equations with two irregular singularities, at zero and at infinity. *** and *** have constructed their holomorphic solutions on C for an explicit class of parameter values and conjectured that they do not exist for other parameter values. They have reduced their conjecture to the second conjecture saying that if an appropriate second similar equation has a polynomial solution, then the first one has no entire solution. They have proved the latter statement under the additional assumption (third conjecture) that f(k,n)(x) (1) 0 for k = (l,....,1), n = (l - 1,....,0) and every x > 0. Our more general result implies all the above conjectures, together with their corollary for the overdamped model of the Josephson junction in superconductivity: the description of adjacency points of phase-lock areas as solutions of explicit analytic equations.

关键词： modified bessel functions double confluent Heun equations entire solutions ordinary differential equations on two-torus rotation number phase-lock areas Josephson effect in superconductivity

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