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The generalized Kaiser–Bessel window function is defined via the modified Bessel function of the first kind and arises frequently in tomographic image reconstruction. In this paper, we study in details the properties of the Kaiser–Bessel distribution, which we define via the symmetric form of the generalized Kaiser–Bessel window function. The Kaiser–Bessel distribution resembles to the Bessel distribution of McKay of the first type, it is a platykurtic or sub-Gaussian distribution, it is not infinitely divisible in the classical sense and it is an extension of the Wigner's semicircle, parabolic and n-sphere distributions, as well as of the ultra-spherical (or hyper-spherical) and power semicircle distributions. We deduce the moments and absolute moments of this distribution and we find its characteristic and moment generating function in two different ways. In addition, we find its cumulative distribution function in three different ways and we deduce a recurrence relation for the moments and absolute moments. Moreover, by using a formula of Ismail and May on quotient of modified Bessel functions of the first kind, we deduce a closed-form expression for the differential entropy. We also prove that the Kaiser–Bessel distribution belongs to the family of log-concave and geometrically concave distributions, and we study in details the monotonicity and convexity properties of the probability density function with respect to the argument and each of the parameters. In the study of the monotonicity with respect to one of the parameters we complement a known result of Gronwall concerning the logarithmic derivative of modified Bessel functions of the first kind. Finally, we also present a modified method of moments to estimate the parameters of the Kaiser–Bessel distribution, and by using the classical rejection method we present two algorithms for sampling independent continuous random variables of Kaiser–Bessel distribution. The paper is closed with conclusions and proposals for future works. [ABSTRACT FROM AUTHOR]

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 asymptotic and numerical behavior of the so-called generalized Marcum function of the second kind is considered. By using some known expansions on the modified Bessel function of the second kind, we deduce the asymptotic expansions for the generalized Marcum function of the second kind for large parameters, and all the expansions are obtained when exactly one parameter is large and others are fixed. We also show numerically that the asymptotic formulas obtained in this paper are good approximations. [ABSTRACT FROM AUTHOR]

In this paper our aim is to find the radii of starlikeness and convexity of the generalized Mittag-Leffler function for three different kinds of normalization by using their Hadamard factorization in such a way that the resulting functions are analytic in the unit disk of the complex plane. The characterization of entire functions from Laguerre-Pólya class and a result of H. Kumar and M.A. Pathan on the reality of the zeros of generalized Mittag-Leffler functions, which origins goes back to Dzhrbashyan, Ostrovskiĭ and Peresyolkova, play important roles in this paper. Moreover, the interlacing properties of the zeros of Mittag-Leffler function and its derivative is also useful in the proof of the main results. By using the Euler-Rayleigh inequalities for the real zeros of the generalized Mittag-Leffler function, we obtain some tight lower and upper bounds for the radii of starlikeness and convexity of order zero. [ABSTRACT FROM AUTHOR]

We derive a closed formula for the determinant of the Hankel matrix whose entries are given by sums of negative powers of the zeros of the regular Coulomb wave function. This new identity applied together with results of Grommer and Chebotarev allows us to prove a Hurwitz-type theorem about the zeros of the regular Coulomb wave function. As a particular case, we obtain a new proof of the classical Hurwitz's theorem from the theory of Bessel functions that is based on algebraic arguments. In addition, several Hankel determinants with entries given by the Rayleigh function and Bernoulli numbers are also evaluated. [ABSTRACT FROM AUTHOR]

In this paper our aim is to show some new inequalities of the Redheffer type for Bessel and modified Bessel functions of the first kind. The key tools in our proofs are some classical results on the monotonicity of quotients of differentiable functions as well as on the monotonicity of quotients of two power series. We also use some known results on the quotients of Bessel and modified Bessel functions of the first kind, and by using the monotonicity of the Dirichlet eta function we prove a sharp inequality for the tangent function. At the end of the paper a conjecture is stated, which may be of interest for further research. [ABSTRACT FROM AUTHOR]

In this paper, we study the local zero behavior of orthogonal polynomials around an algebraic singularity, that is, when the measure of orthogonality is supported on [-1,1] and behaves like h(x)|x-x0|λdx for some x0∈(-1,1), where h(x) is strictly positive and analytic. We shall sharpen the theorem of Yoram Last and Barry Simon and show that the so-called fine zero spacing (which is known for λ=0) unravels in the general case, and the asymptotic behavior of neighbouring zeros around the singularity can be described with the zeros of the function cJλ-12(x)+dJλ+12(x), where Ja(x) denotes the Bessel function of the first kind and order a. Moreover, using Sturm-Liouville theory, we study the behavior of this linear combination of Bessel functions, thus providing estimates for the zeros in question. [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]

In this paper, our aim is to find the radii of starlikeness and convexity of the normalized Wright functions for three different kinds of normalization. The key tools in the proof of our main results are the Mittag-Leffler expansion for Wright function and properties of real zeros of the Wright function and its derivative. In addition, by using the Euler-Rayleigh inequalities we obtain some tight lower and upper bounds for the radii of starlikeness and convexity of order zero for the normalized Wright functions. The main results of the paper are natural extensions of some known results on classical Bessel functions of the first kind. Some open problems are also proposed, which may be of interest for further research. AMS subject classifications: 30C45, 30C15, 33C10 [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]

Turán type inequalities for modified Bessel functions of the first kind are used to deduce some sharp lower and upper bounds for the asymptotic order parameter of the stochastic Kuramoto model. Moreover, approximation from the Lagrange inversion theorem and a rational approximation are given for the asymptotic order parameter. [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]

Some Turán type inequalities for Struve functions of the first kind are deduced by using various methods developed in the case of Bessel functions of the first and second kind. New formulas, like Mittag–Leffler expansion, infinite product representation for Struve functions of the first kind, are obtained, which may be of independent interest. Moreover, some complete monotonicity results and functional inequalities are deduced for Struve functions of the second kind. These results complement naturally the known results for a particular case of Lommel functions of the first kind, and for modified Struve functions of the first and second kind. [ABSTRACT FROM AUTHOR]

In this paper necessary and sufficient conditions are deduced for the close-to-convexity of some special combinations of Bessel functions of the first kind and their derivatives by using a result of Shah and Trimble about transcendental entire functions with univalent derivatives and some newly discovered Mittag-Leffler expansions for Bessel functions of the first kind. [ABSTRACT FROM AUTHOR]

In this note our aim is to present some monotonicity properties of the product of modified Bessel functions of the first and second kind. Certain bounds for the product of modified Bessel functions of the first and second kind are also obtained. These bounds improve and extend known bounds for the product of modified Bessel functions of the first and second kind of order zero. A new Turán type inequality is also given for the product of modified Bessel functions, and some open problems are stated, which may be of interest for further research. [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]

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. [ABSTRACT FROM AUTHOR]

In this paper we investigate the Ulam-Hyers stability of several integral equations with singularity. First we give some results concerning the Ulam-Hyers stability of integral equations with weak singularities. Our approach is also suitable for studying some fractional differential equations. In order to emphasize this aspect we prove that some conditions (5) in S. Abbas, M. Benchohra, Ulam-Hyers stability for the Darboux problem for partial fractional differential and integro-differential equations via Picard operators published in Results Math. 65(2014), 67-79 (respectively condition (3.1) from S. Abbas, M. Benchohra, A. Petruşel, Ulam stability for partial fractional differential inclusions via Picard operators theory, Electron. J. Qual. Theory Differ. Equ., 2014, No. 51, 1-13) can be omitted without losing the validity of the obtained results. In the second part we establish some generalized Ulam-Hyers-Rassias stability results for the Bessel equation and related equations. Our approach is based on fixed point methods and the obtained results are more general than those established by Byungbae Kim and Soon-Mo Jung in Bessel's differential equation and its Hyers-Ulam stability appeared in J. Inequal. Appl., Volume 2007. [ABSTRACT FROM AUTHOR]

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 the first kind, which can be expressed in terms of the hypergeometric function $${}_1F_2$$ . The key tool in our proofs is a result of Shah and Trimble about transcendental entire functions with univalent derivatives. Moreover, a known result of Pólya on entire functions, the infinite product representations and some results on zeros of Bessel, Struve, and Lommel functions of the first kind are used in order to achieve the main results of the paper. [ABSTRACT FROM AUTHOR]

Turán, Mitrinović–Adamović and Wilker type inequalities are deduced for regular Coulomb wave functions. The proofs are based on a Mittag–Leffler expansion for the regular Coulomb wave function, which may be of independent interest. Moreover, some complete monotonicity results concerning the Coulomb zeta functions and some interlacing properties of the zeros of Coulomb wave functions are given. [ABSTRACT FROM AUTHOR]

This paper contains an extension of the Julia--Miller--Mocanu lemma for holomorphic functions defined on the unit disk D endowed by the hyperbolic metric on D. The extension is made by following on the one hand the way of the differential subordinations theory and on the other hand the method of Lagrange multipliers, like in an extension of the Julia-Miller-Mocanu lemma for holomorphic mappings in Cn. [ABSTRACT FROM AUTHOR]

In this paper our aim is to present some subordination and superordination results, using an operator, which involves the normalized form of the generalized Bessel functions of first kind. These results are obtained by investigating some appropriate classes of admissible functions. We obtain also some sandwich-type results, and we point out various known or new special cases of our main results. [ABSTRACT FROM AUTHOR]

We study the power mean inequality for generalized trigonometric and hyperbolic functions with two parameters. The generalized p-trigonometric and ( p, q)-trigonometric functions were introduced by Lindqvist and Takeuchi, respectively. [ABSTRACT FROM AUTHOR]

Lower and upper bounds are deduced for some Jaeger integrals which involve the Bessel functions of the first and second kind. The upper bounds contain some elementary functions as well as incomplete gamma functions, while the lower bounds are expressed also in terms of incomplete gamma functions and are deduced via some known inequalities for Bessel functions of the first and second kinds. [ABSTRACT FROM AUTHOR]

In this paper our aim is to present an elementary proof of an identity of Calogero concerning the zeros of Bessel functions of the first kind. Moreover, by using our elementary approach we present a new identity for the zeros of Bessel functions of the first kind, which in particular reduces to some other new identities. We also show that our method can be applied for the zeros of other special functions, like Struve functions of the first kind, and modified Bessel functions of the second kind. [ABSTRACT FROM AUTHOR]

In this note, we offer some log-concavity properties of certain functions related to Bessel functions of the first kind and modified Bessel functions of the first and second kinds, by solving partially a recent conjecture on the log-convexity/log-concavity properties for modified Bessel functions of the first kind and their derivatives. Moreover, we give an application of the mentioned results by extending two inequalities of van der Corput to Bessel and modified Bessel functions of the first kind. Similar inequalities are proved also for modified Bessel functions of the second kind, as well as for log-concave probability density functions. [ABSTRACT FROM AUTHOR]

In this paper, we determine the radius of convexity for three kinds of normalized Bessel functions of the first kind. In the mentioned cases the normalized Bessel functions are starlike-univalent and convex-univalent, respectively, on the determined disks. The key tools in the proofs of the main results are some new Mittag-Leffler expansions for quotients of Bessel functions of the first kind, special properties of the zeros of Bessel functions of the first kind and their derivative, and the fact that the smallest positive zeros of some Dini functions are less than the first positive zero of the Bessel function of the first kind. Moreover, we find the optimal parameters for which these normalized Bessel functions are convex in the open unit disk. In addition, we disprove a conjecture of Baricz and Ponnusamy concerning the convexity of the Bessel function of the first kind. [ABSTRACT FROM AUTHOR]

Abstract: Various miscellaneous functional inequalities are deduced for the so-called generalized inverse trigonometric and hyperbolic functions. For instance, functional inequalities for sums, difference and quotient of generalized inverse trigonometric and hyperbolic functions are given, as well as some Grünbaum inequalities with the aid of the classical Bernoulli inequality. Moreover, by means of certain already derived bounds, bilateral bounding inequalities are obtained for the generalized hypergeometric Clausen function. [Copyright &y& Elsevier]

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]

We point out an error in the proof of the main result of the paper of Tanabe et al. (Comput Stat 22:145-157, ) concerning a parameter estimation for von Mises-Fisher distributions, we correct the proof of the main result and we present a short alternative proof. [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 paper first we survey the Turán type inequalities and related problems for the Bessel functions of the first kind. Then we extend the known higher order Turán type inequalities for Bessel functions of the first kind to real parameters and we deduce new closed integral representation formulas for the second kind Neumann type series of Bessel functions of the first kind occurring in the study of Turán determinants of Bessel functions of the first kind. At the end of the paper we prove a Turán type inequality for the Bessel functions of the second kind. [ABSTRACT FROM AUTHOR]

It is known that the Struve function H and the modified Struve function L are closely connected to the Bessel function of the first kind J and to the modified Bessel function of the first kind I and possess representations through higher transcendental functions like the generalized hypergeometric F and the Meijer G function. Also, the NIST project and Wolfram formula collection contain a set of Kapteyn type series expansions for L( x). In this paper firstly, we obtain various another type integral representation formulae for L( x) using the technique developed by D. Jankov and the authors. Secondly, we present some summation results for different kind of Neumann, Kapteyn and Schlömilch series built by I( x) and L( x) which are connected by a Sonin-Gubler formula, and by the associated modified Struve differential equation. Finally, solving a Fredholm type convolutional integral equation of the first kind, Bromwich-Wagner line integral expressions are derived for the Bessel function of first kind J and for an associated generalized Schlömilch series. [ABSTRACT FROM AUTHOR]

Two different type integral representation formulae are established for Dini (or Fourier–Dini) series of Bessel functions. The first one is a double definite integral expression derived by using a method successfully applied in the similar questions for Neumann, Kapteyn and Schlömilch series, while the another double indefinite integral formula is concluded by means of the non-homogenous Bessel differential equation. [ABSTRACT FROM PUBLISHER]

Abstract: Let denote the class of analytic functions defined on the unit disk with the normalization , in and satisfy the condition for all and for some real constants and such that . We find conditions on constants and such that functions in are univalent in . As a consequence of our investigation, we present univalence and starlikeness criteria. As applications, we present conditions such that is in , where denotes the suitably normalized form of the generalized Bessel functions of the first kind. [Copyright &y& Elsevier]

Abstract: In this paper our aim is to deduce some Turán type inequalities for -hypergeometric and -confluent hypergeometric functions. In order to obtain the main results we apply the methods developed in the case of classical Kummer and Gauss hypergeometric functions. [Copyright &y& Elsevier]

Some sharp two-sided Turán type inequalities for parabolic cylinder functions and Tricomi confluent hypergeometric functions are deduced. The proofs are based on integral representations for quotients of parabolic cylinder functions and Tricomi confluent hypergeometric functions, which arise in the study of the infinite divisibility of the Fisher-Snedecor F distribution. Moreover, some complete monotonicity results are given concerning Turán determinants of Tricomi confluent hypergeometric functions. These complement and improve some of the results of Ismail and Laforgia (in Constr. Approx. 26:1-9, ). [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 [Integral representation for Neumann series of Bessel functions, Proc. Amer. Math. Soc. 137(7) (2009), pp. 2363–2368] derived a closed-form integral expression for a Neumann series of Bessel functions. In this note, our aim is to establish another kind of integral representations for the Neumann series of Bessel functions of the first kind J ν. [ABSTRACT FROM PUBLISHER]

Abstract: Complete monotonicity, Laguerre and Turán type inequalities are established for the so-called Krätzel function , defined by where and . Moreover, we prove the complete monotonicity of a determinant function of which entries involve the Krätzel function. [Copyright &y& Elsevier]

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]

Abstract: We present several inequalities for where is the incomplete gamma function. One of our theorems states that the inequalities hold for all nonnegative real numbers if and only if and . Here, denotes the power sum of order t. This extends and complements a result published by Ismail and Laforgia in 2006. [Copyright &y& Elsevier]

The main aim of this research note is to establish two different types of integral representation formulae for the Kapteyn series of the first kind. The first one is a double definite integral expression, while the second type includes indefinite integral representation formula. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we consider the generalized Marcum Q-function of order ν >0 real, defined bywhere a >0, b ⩾0 and I ν stands for the modified Bessel function of the first kind. Our aim is to extend some results on the (first order) Marcum Q-function to the generalized Marcum Q-function in order to deduce some new lower and upper bounds. Moreover, we show that the proposed bounds are very tight for the generalized Marcum Q-function of integer order, and we deduce some new inequalities for the more general case of real order. The chief tools in our proofs are some monotonicity properties of certain functions involving the modified Bessel function of the first kind, which are based on a criterion for the monotonicity of the quotient of two Maclaurin series. [ABSTRACT FROM AUTHOR]

In this paper our aim is to deduce some sharp Turán type inequalities for modified Bessel functions of the first and second kinds. Our proofs are based on explicit formulas for the Turánians of the modified Bessel functions of the first and second kinds and on a formula which is related to the infinite divisibility of the Student t-distribution. [ABSTRACT FROM AUTHOR]

In this paper, we give sufficient conditions for the parameters of the normalized form of the generalized Bessel functions to be convex and starlike in the open unit disk. As an application of our main results, we solve a recent open problem concerning a subordination property of Bessel functions with different parameters. Moreover, we present a new inequality for the Euler gamma function, which we apply in order to have tight bounds for the generalized and normalized Bessel function of the first kind. [ABSTRACT FROM AUTHOR]

Abstract: In this note our aim is to deduce some sufficient conditions for integral operators involving Bessel functions of the first kind to be univalent in the open unit disk. The key tools in our proofs are the generalized versions of the well-known Ahlfors’ and Becker’s univalence criteria and some inequalities for the normalized Bessel functions of the first kind. [Copyright &y& Elsevier]