Related papers: Partial fraction expansions and zeros of Hankel tr…
The real and complex zeros of some special entire functions such as Wright, hyper-Bessel, and a special case of generalized hypergeometric functions are studied by using some classical results of Laguerre, Obreschkhoff, P\'olya and Runckel.…
The technique of P\'{o}lya-Hurwitz of partial fractions is implemented to investigate the zeros of finite $q$-Hankel transforms, which are defined in terms of the third $q$-Bessel function of Jackson. The new approach, which is a…
We show connections between a special type of addition formulas and a theorem of Stieltjes and Rogers. We use different techniques to derive the desirable addition formulas. We apply our approach to derive special addition theorems for…
In this paper our aim is to deduce some sufficient (and necessary) conditions for the close-to-convexity of some special functions and their derivatives, like Bessel functions, Struve functions, and a particular case of Lommel functions of…
We study Hankel transforms of sequences, where the transform elements are members of the set {-1,0,1}. We relate these Hankel transforms to special continued fraction expansions. In particular, we posit a conjecture relating the…
Fractional vector calculus is discussed in the spherical coordinate framework. A variation of the Legendre equation and fractional Bessel equation are solved by series expansion and numerically. Finally, we generalize the hypergeometric…
The Hankel transform H_n[f(x)](q) = int_0^infinity xf(x)J_n(qx)dx is studied for integer n>=-1 and positive parameter q. It is proved that the Hankel transform is given by uniformly and absolutely convergent series in reciprocal powers of…
We will prove an analogue of Landau's necessary conditions [Necessary density conditions for sampling and interpolation of certain entire functions, Acta Math. 117 (1967).] for spaces of functions whose Hankel transform is supported in a…
It is shown how sums of squares of real valued functions can be used to give new proofs of the reality of the zeros of the Bessel functions $J_\alpha (z)$ when $\alpha \ge -1,$ confluent hypergeometric functions ${}_0F_1(c\/; z)$ when $c>0$…
In consideration of the integral transform whose kernel arises as an oscillatory solution of certain second-order linear differential equation, its positivity is investigated on the basis of Sturm's theory. As applications, positivity…
Elementary transformations of equations $A\psi=\lambda\psi$ are considered. The invertibility condition (Theorem 1) is established and similar transformations of Riccati equations in the case of second order differential operator $A$ are…
We study the distribution of the zeros of functions of the form $f(s)=h(s) \pm h(2a-s)$, where $h(s)$ is a meromorphic function, real on the real line, $a$ a real number. One of our results establishes sufficient conditions under which all…
We study convexity properties of the zeros of some special functions that follow from the convexity theorem of Sturm. We prove results on the intervals of convexity for the zeros of Laguerre, Jacobi and ultraspherical polynomials, as well…
The article discusses the fractional powers of the Bessel operator and their numerical implementation. An extensive literature is devoted to the study of fractional powers of the Laplace operator and their applications. Such degrees are…
This is the first installment in a series of papers devoted to examining certain aspects of the asymptotic value distribution and distribution of zeros manifested by members of a broad class of linear combinations of L-functions in the…
We derive a convergent expansion of the generalized hypergeometric function ${}_{p-1}F_p$ in terms of the Bessel functions ${}_{0}F_1$ that holds uniformly with respect to the argument in any horizontal strip of the complex plane. We…
In this paper, we aim to present new extensions of incomplete gamma, beta, Gauss hypergeometric, confluent hypergeometric function and Appell-Lauricella hypergeometric functions, by using the extended Bessel function due to Boudjelkha [4].…
The generalization, similarly to exponential multivariate bases in the Fourier transform, of the Bessel functions to many dimensions is offered. Analogously to the Fourier transform property under the differentiation, the similar Hankel…
The essentials of fractional calculus according to different approaches that can be useful for our applications in the theory of probability and stochastic processes are established. In addition to this, from this fractional integral one…
Fractional calculus is the calculus of differentiation and integration of non-integer orders. In a recently paper (Annals of Physics 323 (2008) 2756-2778), the Fundamental Theorem of Fractional Calculus is highlighted. Based on this…