On an identity for zeros of Bessel functions

In this paper our aim is to present an elementary proof of an identity of Calogero concerning the zeros of Bessel functions of the first kind. Moreover, by using our elementary approach we present a new identity for the zeros of Bessel functions of the first kind, which in particular reduces to some other new identities. We also show that our method can be applied for the zeros of other special functions, like Struve functions of the first kind, and modified Bessel functions of the second kind.     

Functional inequalities involving Bessel and modified Bessel functions of the first kind

In this paper, we extend some known elementary trigonometric inequalities, and their hyperbolic analogues to Bessel and modified Bessel functions of the first kind. In order to prove our main results, we present some monotonicity and convexity properties of some functions involving Bessel and modified Bessel functions of the first kind. We also deduce some Turán and Lazarević-type inequalities for the confluent hypergeometric functions.     

Bounds for the product of modified Bessel functions

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 Turán 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.     

Redheffer type bounds for Bessel and modified Bessel functions of the first kind

In this paper our aim is to show some new inequalities of the Redheffer type for Bessel and modified Bessel functions of the first kind. The key tools in our proofs are some classical results on the monotonicity of quotients of differentiable functions as well as on the monotonicity of quotients of two power series. We also use some known results on the quotients of Bessel and modified Bessel functions of the first kind, and by using the monotonicity of the Dirichlet eta function we prove a sharp inequality for the tangent function. At the end of the paper a conjecture is stated, which may be of interest for further research.     

Modified Dini functions: monotonicity patterns and functional inequalities

We deduce some new functional inequalities, like Turán type inequalities, Redheffer type inequalities, and a Mittag-Leffler expansion for a special combination of modified Bessel functions of the first kind, called modified Dini functions. Moreover, we show the complete monotonicity of a quotient of modified Dini functions by involving a new continuous infinitely divisible probability distribution. The key tool in our proofs is a recently developed infinite product representation for a special combination of Bessel functions of the first kind, which was very useful in determining the radius of convexity of some normalized Bessel functions of the first kind.     

On a Sum of Modified Bessel Functions

In this paper we consider a sum of modified Bessel functions of the first kind of which particular case is used in the study of Kanter’s sharp modified Bessel function bound for concentrations of some sums of independent symmetric random vectors. We present some monotonicity and convexity properties for that sum of modified Bessel functions of the first kind, as well as some Turán type inequalities, lower and upper bounds. Moreover, we point out an error in Kanter’s paper (J Multivariate Anal 6:222–236, 1976).     

Close‐to‐convexity of normalized Dini functions

In this paper necessary and sufficient conditions are deduced for the close‐to‐convexity of some special combinations of Bessel functions of the first kind and their derivatives by using a result of Shah and Trimble about transcendental entire functions with univalent derivatives and some newly discovered Mittag–Leffler expansions for Bessel functions of the first kind.     

Functional inequalities for modified Bessel functions

In this paper, our aim is to show some mean value inequalities for the modified Bessel functions of the first and second kind. Our proofs are based on some bounds for the logarithmic derivatives of these functions, which are in fact equivalent to the corresponding Turán-type inequalities for these functions. As an application of the results concerning the modified Bessel function of the second kind, we prove that the cumulative distribution function of the gamma–gamma distribution is log-concave. At the end of this paper, several open problems are posed, which may be of interest for further research.     

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 of the second kind. As applications, we prove the logarithmically monotonicity for a class of functions involving the modified Bessel functions of second kind and we established new inequalities for this function.     

Numerical solution of the Bagley–Torvik equation by the Bessel collocation method

In this article, a numerical technique is presented for the approximate solution of the Bagley–Torvik equation, which is a class of fractional differential equations. The basic idea of this method is to obtain the approximate solution in a generalized form of the Bessel functions of the first kind. For this purpose, by using the collocation points, the matrix operations and a generalization of the Bessel functions of the first kind, this technique transforms the Bagley–Torvik equation into a system of the linear algebraic equations. Hence, by solving this system, the unknown Bessel coefficients are computed. The reliability and efficiency of the proposed scheme are demonstrated by some numerical examples. Copyright © 2012 John Wiley & Sons, Ltd.     

Zeros of combinations of Bessel functions and the mean charge of graphene nanodots

We establish some properties of the zeros of sums and differences of contiguous Bessel functions of the first kind. As a by-product, we also prove that the zeros of the derivatives of Bessel functions of the first kind of different orders are interlaced the same way as the zeros of the Bessel functions themselves. As a physical motivation, we consider gated graphene nanodots subject to Berry–Mondragon boundary conditions. We determine the allowed energy levels and calculate the mean charge at zero temperature. We discuss its dependence on the gate (chemical) potential in detail and also comment on the effect of temperature.     

Inequalities involving modified Bessel functions of the first kind II

The intrinsic properties, including logarithmic convexity (concavity), of the modified Bessel functions of the first kind and some other related functions are obtained. Several inequalities involving functions under discussion are established.     

Luigi Gatteschi’s work on asymptotics of special functions and their zeros

A good portion of Gatteschi’s research publications—about 65%—is devoted to asymptotics of special functions and their zeros. Most prominently among the special functions studied figure classical orthogonal polynomials, notably Jacobi polynomials and their special cases, Laguerre polynomials, and Hermite polynomials by implication. Other important classes of special functions dealt with are Bessel functions of the first and second kind, Airy functions, and confluent hypergeometric functions, both in Tricomi’s and Whittaker’s form. This work is reviewed here, and organized along methodological lines.     

Cross-product of Bessel functions: Monotonicity patterns and functional inequalities

In this paper, we study the Dini functions and the cross-product of Bessel functions. Moreover, we are interested on the monotonicity patterns for the cross-product of Bessel and modified Bessel functions. In addition, we deduce Redheffer-type inequalities, and the interlacing property of the zeros of Dini functions and the cross-product of Bessel and modified Bessel functions. Bounds for logarithmic derivatives of these functions are also derived. The key tools in our proofs are some recently developed infinite product representations for Dini functions and cross-product of Bessel functions.     

Generalized Bessel functions: Theory and their applications

This paper presents 2 new classes of the Bessel functions on a compact domain [0,T] as generalized‐tempered Bessel functions of the first‐ and second‐kind which are denoted by GTBFs‐1 and GTBFs‐2. Two special cases corresponding to the GTBFs‐1 and GTBFs‐2 are considered. We first prove that these functions are as the solutions of 2 linear differential operators and then show that these operators are self‐adjoint on suitable domains. Some interesting properties of these sets of functions such as orthogonality, completeness, fractional derivatives and integrals, recursive relations, asymptotic formulas, and so on are proved in detail. Finally, these functions are performed to approximate some functions and also to solve 3 practical differential equations of fractionalorders.     

New formulae for higher order derivatives and applications

We present new formulae (the Slevinsky–Safouhi formulae I and II) for the analytical development of higher order derivatives. These formulae, which are analytic and exact, represent the kth derivative as a discrete sum of only k+1 terms. Involved in the expression for the kth derivative are coefficients of the terms in the summation. These coefficients can be computed recursively and they are not subject to any computational instability. As examples of applications, we develop higher order derivatives of Legendre functions, Chebyshev polynomials of the first kind, Hermite functions and Bessel functions. We also show the general classes of functions to which our new formula is applicable and show how our formula can be applied to certain classes of differential equations. We also presented an application of the formulae of higher order derivatives combined with extrapolation methods in the numerical integration of spherical Bessel integral functions.     

Calculation of Bessel functions by using continued fractions

We propose a new method for the calculation of Bessel functions of the first kind of integral order. By using the Laplace transformation, we solve a linear differential equation that defines the generating function for the Bessel functions expressed in terms of continued fractions.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 12, pp. 1704–1705, December, 1995.     

Approximate formulas for moderately small eikonal amplitudes

We consider the eikonal approximation for moderately small scattering amplitudes. To find numerical estimates of these approximations, we derive formulas that contain no Bessel functions and consequently no rapidly oscillating integrands. To obtain these formulas, we study improper integrals of the first kind containing products of the Bessel functions J₀(z). We generalize the expression with four functions J₀(z) and also find expressions for the integrals with the product of five and six Bessel functions. We generalize a known formula for the improper integral with two functions J_υ (az) to the case with noninteger υ and complex a.     

Function evaluation using expansions in Bessel functions of the first kind

An iterative method is proposed for evaluation of fonctions that are expandable in series of Bessel functions of the first kind. The Bessel functions are evaluated by Miller's method, avoiding the need to determine their exact values. As an example, we describe algorithms for evaluation of the integral sine and the normal probability integral with an accuracy of to 10–12 significant digits.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 61, pp. 37–42, 1987.     

On the summation of Schlömilch's series

ABSTRACT

Schlömilch's series is named after the German mathematician Oscar Xavier Schlömilch, who derived it in 1857 as a Fourier series type expansion in terms of the Bessel function of the first kind. However, except for Bessel functions, here we consider an expansion in terms of Struve functions or Bessel and Struve integrals as well. The method for obtaining a sum of Schlömilch's series in terms of the Bessel or Struve functions is based on the summation of trigonometric series, which can be represented in terms of the Riemann zeta and related functions of reciprocal powers and in certain cases can be brought in the closed form, meaning that the infinite series are represented by finite sums. By using Krylov's method we obtain the convergence acceleration of the trigonometric series.