Search

In this paper, by using an integral representation of Ismail and Kelker for the quotient of Tricomi hypergeometric functions, we investigate the infinite divisibility and self-decomposability of the recently defined four-parameter lifetime Whittaker distribution, which is a natural extension of the classical gamma, exponential, chi-square, generalized Lindley, Lindley, beta prime, and Lomax distributions. We also show that the Whittaker distribution belongs to the class of hyperbolically completely monotone distributions and generalized gamma convolutions, and it is a super-Gaussian distribution. By using some results for the moments of the Whittaker distribution, we also deduce some Turán type inequalities for the Whittaker functions of the second kind and as an application we show that the effective variance of the Whittaker distribution is bounded from below. [ABSTRACT FROM AUTHOR]

Some well-known results of Grönwall on logarithmic derivative of modified Bessel functions of the first kind concerning exponential bounds are extended to Whittaker functions of the first and second kind M_{\kappa,\mu } and W_{\kappa,\mu }. Moreover, a complete monotonicity result is proved for the logarithmic derivative of the Whittaker function W_{\kappa,\mu }, and some monotonicity results with respect to the parameters and argument are shown for the logarithmic derivative of M_{\kappa,\mu }. The results extend and complement the known results in the literature about modified Bessel functions of the first and second kind. [ABSTRACT FROM AUTHOR]

In this paper, we study some geometric properties of a class of analytic functions which is defined from the J-fraction expansion of the ratio {zf'(z)}/{f(z)}. We find the disk domain which is mapped into a starlike domain by these functions. Moreover, we study similar results for two different normalized forms of regular Coulomb wave functions and a normalized Bessel function of the first kind by using continued fractions expansions. [ABSTRACT FROM AUTHOR]

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 obtained results extend the known theorem of Hurwitz on the exact number of nonreal zeros of Bessel functions of the first kind. Moreover, results on zeros of derivatives of Bessel functions and the crossproduct of Bessel functions are also given, which are related to some recent open problems. [ABSTRACT FROM AUTHOR]

We prove that for 𝑣 > n − 1 all zeros of the nth derivative of the Bessel function of the first kind J𝑣 are real. Moreover, we show that the positive zeros of the nth and (n + 1)th derivative of the Bessel function of the first kind J𝑣 are interlacing when 𝑣 ≥ n and n is a natural number or zero. Our methods include the Weierstrassian representation of the nth derivative, properties of the Laguerre-Pólya class of entire functions, and the Laguerre inequalities. Some similar results for the zeros of the first and second derivatives of the Struve function of the first kind 𝐇𝑣 are also proved. The main results obtained in this paper generalize and complement some classical results on the zeros of Bessel and Struve functions of the first kind. Some open problems related to Hurwitz’s theorem on the zeros of Bessel functions are also proposed. [ABSTRACT FROM AUTHOR]

The Lambert W function, giving the solutions of a simple transcendental equation, has become a famous function and arises in many applications in combinatorics, physics, or population dyamics just to mention a few. In this paper we construct and study in great detail a generalization of the Lambert W which involves some special polynomials and even combinatorial aspects. [ABSTRACT FROM AUTHOR]

Motivated by a problem on conditions for the existence of clines in genetics, we show that the positive zeros of a cross-product of the regular Coulomb wave function and the Tricomi hypergeometric function are increasing with respect to one of the parameters. In particular, this implies that the eigenvalues of a certain boundary value problem are increasing with the dimension. [ABSTRACT FROM AUTHOR]

Geometric properties of the classical Lommel and Struve functions, both of the first kind, 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 radius of starlikeness precisely. [ABSTRACT FROM AUTHOR]

In this paper our aim is to present some Landen inequalities for Gaussian hypergeometric functions, confluent hypergeometric functions, generalized Bessel functions and general power series. Our main results complement and generalize some known results in the literature. [ABSTRACT FROM AUTHOR]

In this note our aim is to determine the radius of starlikeness of the normalized Bessel functions of the first kind for three different kinds of normalization. The key tool in the proof of our main result is the Mittag-Leffler expansion for Bessel functions of the first kind and the fact that, according to Ismail and Muldoon, the smallest positive zeros of some Dini functions are less than the first positive zero of the Bessel function of the first kind. [ABSTRACT FROM AUTHOR]

In this note our aim is to point out that certain inequalities for modified Bessel functions of the first and second kind, deduced recently by Laforgia and Natalini, are in fact equivalent to the corresponding Turán type inequalities for these functions. Moreover, we present some new Turán type inequalities for the aforementioned functions and we show that their product is decreasing as a function of the order, which has an application in the study of stability of radially symmetric solutions in a generalized FitzHugh-Nagumo equation in two spatial dimensions. At the end of this note an open problem is posed, which may be of interest for further research. [ABSTRACT FROM AUTHOR]

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 of the first kind Jv . In this paper our aim is to establish analogous integral representations for the Neumann-type series of modified Bessel functions of the first kind Iv and for Bessel functions of the second kind Yv, Kv, and to give links for the same question for the Hankel functions Hv(1), Hv(2) [ABSTRACT FROM AUTHOR]