© Hindawi Publishing Corp. THE ROOTS OF THE THIRD JACKSON q-BESSEL FUNCTION

We derive analytic bounds for the zeros of the third Jackson 
 q -Bessel function J v ( 3 ) ( z ; q ) .

(1. 1) This function is also known as the Hahn-Exton q-Bessel function [8,9].The notation 1 Φ 1 in (1.1) is the standard in use for q-hypergeometric series [4].The function J (3) ν (z; q) satisfies a linear q-difference equation and it is known that J (3) ν (z; q) has an infinite number of simple real zeros [8].In this paper, we will give lower and upper bounds for these zeros.The roots of these functions are of interest for several reasons.Firstly, it is intrinsically interesting to provide information about the roots of a function such as (1.1), which is an entire function of order zero.Also, the roots of J (2) ν (z; q) and J (3) ν (z; q) figure prominently in expansions in terms of "q-Fourier series" [2,3].Lastly, if we denote the roots of J (3) ν (z; q) by j (3) n,ν , then the mass points of the orthogonality measure for a qanalog of Lommel polynomials are located at the points 1/j (3) n,ν .Furthermore, although the function defined in (1.1) is of a simpler character than the remaining Jackson q-Bessel functions, it is hoped that the results given here for J (3) ν (z; q) may be extended in the future to

The roots of J
(3) ν (z; q).We prove two lemmas stating the existence of an odd number of roots in a certain interval and then we prove that J (3) ν (z; q) has only one root in such an interval.
(2.3)This representation will be critical in the proof of the next two lemmas.
Remark 2.6.Lemmas 2.1 and 2.2 and Theorem 2.3 state that the roots w (ν) k (q) satisfy the inequalities (2.29) These bounds are quite accurate.This is evident if we estimate the length of the interval containing the roots.A somewhat tedious calculation with Taylor series shows that q −m/2 − q −m/2+α (ν)  m (q) = q m/2+ν O(1). (2.30) Clearly, for fixed q satisfying the conditions of the theorem, the bounds become increasingly accurate as either k or ν increases.

Call for Papers
This subject has been extensively studied in the past years for one-, two-, and three-dimensional space.Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon.Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall.This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles.His original model was then modified and considered under different approaches and using many versions.Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos).We intend to publish in this special issue papers reporting research on time-dependent billiards.The topic includes both conservative and dissipative dynamics.Papers discussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.
To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned.Mathematical papers regarding the topics above are also welcome.