Jackson q-Bessel Function Research Articles

The q-Lidstone series involving q-Bernoulli and q-Euler polynomials generated by the third Jackson q-Bessel function

The q-Lidstone series involving q-Bernoulli and q-Euler polynomials generated by the third Jackson q-Bessel function

A q-Difference Equation and Fourier Series Expansions of q-Lidstone Polynomials

In this paper, we present the q-Lidstone polynomials which are q-Bernoulli polynomials generated by the third Jackson q-Bessel function, based on the Green’s function of a certain q-difference equation. Also, we provide the q-Fourier series expansions of these polynomials and derive some results related to them.

Spectral analysis of two doubly infinite Jacobi matrices with exponential entries

We provide a complete spectral analysis of all self-adjoint operators acting on ℓ2(Z) which are associated with two doubly infinite Jacobi matrices with entries given byq−n+1δm,n−1+q−nδm,n+1 andδm,n−1+αq−nδm,n+δm,n+1, respectively, where q∈(0,1) and α∈R. As an application, we derive orthogonality relations for the Ramanujan entire function and the third Jackson q-Bessel function.

Uniform convergence of basic Fourier\u2013Bessel series on a q-linear grid

We study Fourier–Bessel series on a q-linear grid, defined as expansions in complete q-orthogonal systems constructed with the third Jackson q-Bessel function, and obtain sufficient conditions for uniform convergence. The convergence results are illustrated with specific examples of expansions in q-Fourier–Bessel series.

A few properties of the third Jackson q-Bessel function

The Hahn–Exton (third Jackson) q-Bessel function attracts the attention of many researchers in mathematics. We add a few more properties of this important function, such as some identities and upper bounds involving this function, in relation with the coefficients of the corresponding basic Fourier–Bessel expansions.

On certain triple q-integral equations involving the third Jackson q-Bessel functions as kernel

In this paper, we consider two different systems of triple q-integral equations, where the kernel is a third Jackson q-Bessel function. The solution of the first system is reduced to two simultaneous Fredholm q-integral equations of the second kind. We solve the second system by using the solution of specific dual q-integral equations. Some examples are included.

Radii of starlikeness and convexity of some q-Bessel functions

Geometric properties of the Jackson and Hahn–Exton q-Bessel functions are studied. For each of them, three different normalizations are applied in such a way that the resulting functions are analytic in the unit disk of the complex plane. For each of the six functions we determine the radii of starlikeness and convexity precisely by using their Hadamard factorization. These are q-generalizations of some known results for Bessel functions of the first kind. The characterization of entire functions from the Laguerre–Pólya class via hyperbolic polynomials plays an important role in this paper. Moreover, the interlacing property of the zeros of Jackson and Hahn–Exton q-Bessel functions and their derivatives is also useful in the proof of the main results. We also deduce a necessary and sufficient condition for the close-to-convexity of a normalized Jackson q-Bessel function and its derivatives. Some open problems are proposed at the end of the paper.

Positivity of the generalized translation associated with the q-Hankel transform and applications

A new formulation of the Graf-type addition formula related to the third Jackson q-Bessel function gives a solution for the problem of the positivity of the generalized q-translation operator associated with the q-Hankel transform. Next some applications in q-theory are treated, for instance the relationship between the q-Bessel- positive and negative-definite functions. We also show how the positivity of the q-Bessel translation operator plays a central role in q-Fourier analysis, namely in the study of Markov operators in the q-context. The paper concludes with the nonnegative product linearization of the q−2-Lommel polynomials.

Inequalities related to the third Jackson q-Bessel function of order zero

In Bettaibi and Bouzeffour (J. Math. Anal. Appl. 342:1203-1219, 2008), some properties of the third Jackson q-Bessel function of order zero were established. This paper is devoted to studying the q-convolution product by using a q-integral representation of the related q-translation. The central part of this work is first to study the related q-heat semi-group and its hypercontractivity and second to specify the q-analogue of the Wiener algebra.

A q-linear analogue of the plane wave expansion

We obtain a q-linear analogue of Gegenbauerʼs expansion of the plane wave. It is expanded in terms of the little q-Gegenbauer polynomials and the third Jackson q-Bessel function. The result is obtained by using a method based on bilinear biorthogonal expansions.

Positivity of the Generalized Translation Associated with the q-Hankel Transform

Using a new formulation of Graf’s addition formula related to the third Jackson q-Bessel function, we study the positivity of the generalized q-translation operator associated with the q-Hankel transform.

Plancherel–Rotach asymptotics for certain basic hypergeometric series

In this work we study the chaotic and periodic asymptotics for the confluent basic hypergeometric series. For a fixed q ∈ ( 0 , 1 ) , the asymptotics for Euler's q-exponential, q-Gamma function Γ q ( x ) , q-Airy function of K. Kajiwara, T. Masuda, M. Noumi, Y. Ohta and Y. Yamada, Ramanujan function ( q-Airy function), Jackson's q-Bessel function of second kind, Ismail–Masson orthogonal polynomials ( q −1 -Hermite polynomials), Stieltjes–Wigert polynomials, q-Laguerre polynomials could be derived as special cases.

On the zeros of basic finite Hankel transforms

In this article we prove that the basic finite Hankel transform whose kernel is the third-type Jackson q-Bessel function has only infinitely many real and simple zeros, provided that q satisfies a condition additional to the standard one. We also study the asymptotic behavior of the zeros. The obtained results are applied to investigate the zeros of q-Bessel functions as well as the zeros of q-trigonometric functions. A basic analog of a theorem of G. Pólya (1918) on the zeros of sine and cosine transformations is also given.

An integral representation and some transformation properties of q-Bessel functions

A Poisson-type integral representation for Jackson's q-Bessel function is obtained by using Askey and Wilson's q-beta integral and Nassrallah and Rahman's integral formula for an 8 ϑ 7 series. This representation along with some transformtion formulas for basic hypergeometric series help express the q-Bessel functions as a 3 ϑ 2 series in base q, a 2 ϑ 2 series in base q 2 and a 2 ϑ 1 series in base q q1 2 , where 0 < q < 1. As an application, q-analogues are found for Gegenbauer's degenerate addition formulas for Bessel functions.