Interlacing of positive real zeros of Bessel functions

We unify the three distinct inequality sequences (Abramowitz and Stegun (1972) [1, 9.5.2]) of positive real zeros of Bessel functions into a single one.     

Elementary approximations for zeros of Bessel functions

Let j_vk, y_vk and c_vk denote the kth positive zeros of the Bessel functions J_v(x), Y_v(x) and of the general cylinder function C_v(x) = cos αJ_v(x)?sin αY_v(x), 0 ? α < π, respectively. In this paper we extend to c_vk, k = 2, 3,..., some linear inequalities presently known only for j_vk. In the case of the zeros y_vk we are able to extend these inequalities also to k = 1. Finally in the case of the first positive zero j_v1 we compare the linear enequalities given in [9] with some other known inequalities.     

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.     

On integral representation of Bessel function of the first kind

A first kind Fredholm integral equation with nondegenerate kernel is given, which particular solution is the Bessel function of the first kind. This equation is solved by means of Mellin transform pair.     

Continuous ranking of zeros of special functions

We reexamine and continue the work of J. Vosmansky [J. Vosmanský, Zeros of solutions of linear differential equations as continuous functions of the parameter k, in: J. Wiener, J.K. Hale (Eds.), Partial Differential Equations, Proceedings of Conference, Edinburg, TX, 1991, in: Pitman Res. Notes Math. Ser., vol. 273, 1992, pp. 253-257] on the concept of continuous ranking of zeros of certain special functions from the point of view of the transformation theory of second-order linear differential equations. This leads to results on higher monotonicity of such zeros with respect to the rank and to the evaluation of some definite integrals. The applications are to Airy, Bessel and Hermite functions.     

Inequalities for modified Bessel functions and their integrals

Simple inequalities for some integrals involving the modified Bessel functions _Iν(x)$I_{\nu }(x)$ and _Kν(x)$K_{\nu }(x)$ are established. We also obtain a monotonicity result for _Kν(x)$K_{\nu }(x)$ and a new lower bound, that involves gamma functions, for _K0(x)$K_0(x)$.     

On a conjecture regarding the extrema of Bessel functions and its generalization

It was conjectured by Á. Elbert in J. Comput. Appl. Math. 133 (2001) 65-83 that, given two consecutive real zeros of a Bessel function of order ν, j_ν,κ and j_ν,κ+1, the zero of the derivative between such two zeros j_ν,κ′ satisfies . We prove that this inequality holds for any Bessel function of any real order. In addition to these lower bounds, upper bounds are obtained. In this way we bracket the zeros of the derivative. It is discussed how similar relations can be obtained for other special functions which are solutions of a second order ODE; in particular, the case of the zeros of is considered.     

k-途径正则有向图

In this paper, we define a class of strongly connected digraph, called the k-walk- regular digraph, study some properties of it, provide its some algebraic characterization and point out that the 0-walk-regular digraph is the same as the walk-regular digraph discussed by Liu and Lin in 2010 and the D-walk-regular digraph is identical with the weakly distance-regular digraph defined by Comellas et al in 2004.     

Quadrature formulae using zeros of Bessel functions as nodes

A gaussian type quadrature formula, where the nodes are the zeros of Bessel functions of the first kind of order (), was recently proved for entire functions of exponential type. Here we relax the restriction on as well as on the function. Some applications are also given.

     

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.     

On the coefficients of Neumann series of Bessel functions

Recently Pogány and Süli (Proc. Amer. Math. Soc. 137 (7) (2009) 2363-2368) derived a closed-form integral expression for Neumann series of Bessel functions. In this note we precisely characterize the class of functions α that generate the integral representation of a Neumann series of Bessel functions in the sense that the restriction αN_|=(αn) of a function α to the set N of all positive integers is the sequence of coefficients of the initial Neumann series.     

Bounds on differences of adjacent zeros of Bessel functions and iterative relations between consecutive zeros

Bounds for the distance between adjacent zeros of cylinder functions are given; and are such that ; stands for the th positive zero of the cylinder (Bessel) function , , .

These bounds, together with the application of modified (global) Newton methods based on the monotonic functions and , give rise to forward ( ) and backward ( ) iterative relations between consecutive zeros of cylinder functions.

The problem of finding all the positive real zeros of Bessel functions for any real and inside an interval , 0$">, is solved in a simple way.

     

Two-sided bounds uniform in the real argument and the index for modified Bessel functions

Bounds uniform in the real argument and the index for the functionsa _ν (x)=xI′ _ν (x)/I′ _ν (x) andb _ν (x)=xK′ _ν (x)/K _ν (x), as well as for the modified Bessel functionsI _ν(x) andK _ν(x), are established in the quadrantx>0, ν≥0, except for some neighborhoods of the pointx=0, ν=0. Translated fromMatematicheskie Zametki, Vol. 65, No. 5, pp. 681–692, May, 1999.     

Discrete Bessel functions and partial difference equations

We introduce a new class of discrete Bessel functions and discrete modified Bessel functions of integer order. After obtaining some of their basic properties, we show that these functions lead to fundamental solutions of the discrete wave equation and discrete diffusion equation.     

Hardy-type theorem for orthogonal functions with respect to their zeros. The Jacobi weight case

Motivated by the G.H. Hardy's 1939 results [G.H. Hardy, Notes on special systems of orthogonal functions II: On functions orthogonal with respect to their own zeros, J. London Math. Soc. 14 (1939) 37-44] on functions orthogonal with respect to their real zeros λn, , we will consider, under the same general conditions imposed by Hardy, functions satisfying an orthogonality with respect to their zeros with Jacobi weights on the interval (0,1), that is, the functions f(z)=zνF(z), ν∈R, where F is entire and