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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]

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]

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]

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]

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 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 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]

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]

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 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]

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]

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: 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]

Some new inequalities for quotients of modified Bessel functions of the first and second kinds are deduced. Moreover, some developments on bounds for modified Bessel functions of the first and second kinds, higher-order monotonicity properties of these functions and applications to a special function that arises in finite elasticity, are summarized. The key tool in our proofs is a frequently used criterion for the monotonicity of the quotient of two Maclaurin series. [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]

Abstract: In this paper we study the generalized Marcum Q-function of order real, defined by where , and stands for the modified Bessel function of the first kind. Our aim is to improve and extend some recent results of Wang to the generalized Marcum Q-function in order to deduce some sharp lower and upper bounds. In both cases and we give the best possible upper bound for . The key tools in our proofs are some monotonicity properties of certain functions involving the modified Bessel function of the first kind. These monotonicity properties are deduced from some results on modified Bessel functions, which have been used in wave mechanics and finite elasticity. [Copyright &y& Elsevier]

In this paper, we study the generalized Marcum Q-function Qv(a,b), where a,v > 0 and b ≥ 0. Our aim is to extend the results of Corazza and Ferrari (IEEE Trans. Inf. Theory, vol. 48, pp. 3003-3008, 2002) to the generalized Marcum Q-function in order to deduce some new tight lower and upper bounds. The key tools in our proofs are some monotonicity properties of certain functions involving the modified Bessel function of the first kind and some classical inequalities, i.e., the Cauchy-Buniakowski-Schwarz and Chebyshev integral inequalities. These bounds are shown to be very tight for large b, i.e., the relative errors of our bounds converge to zero as b increases. Both theoretical analysis and numerical results are provided to show the tightness of our bounds. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we consider the generalized Marcum Q-function of order real, defined bywhere , stands for the modified Bessel function of the first kind and the right hand side of the above equation is replaced by its limiting value when . Our aim is to prove that the function is strictly increasing on for each , , and to deduce some interesting inequalities for the function . Moreover, we present a somewhat new viewpoint of the generalized Marcum Q-function, by showing that satisfies the new-is-better-than-used (nbu) property, which arises in economic theory. [Copyright &y& Elsevier]

Abstract: The intrinsic properties, including logarithmic convexity (concavity), of the modified Bessel functions of the first kind and some other related functions are obtained. Several inequalities involving functions under discussion are established. [Copyright &y& Elsevier]

Abstract: This paper is motivated by an open problem of Anderson, Vamanamurthy, and Vuorinen. We prove a chain of inequalities for hypergeometric functions, generalized and normalized Bessel functions, Kummer functions and for some general power series. Some particular cases and refinements are given. [Copyright &y& Elsevier]

The article presents a study on trigonometric inequalities and its extension to Bessel functions. The aim of the study is to stretch some A. Oppenheim's inequalities to Bessel functions. The authors deduce the hyperbolic analogue of these trigonometric inequalities for the said extension. In 1958, Oppenheim and Carver provided a partial solution to Oppenheim's problems, while L. Zhu has recently solved the said problem completely. Also, Á Baricz extended the Carver solution to Bessel functions.

Abstract: Let be the generalized and normalized Bessel function depending on parameters and let , . Motivated by an open problem of Anderson, Vamanamurthy, and Vuorinen we prove that for all for certain conditions on the parameters . [Copyright &y& Elsevier]