We investigate the asymptotic expansion of integrals analogous to Ball's integral \[\int_0^\infty \left(\frac{Γ(1+ν)|J_ν(x)|}{(x/2)^ν}\right)^{\!n}dx\] for large $n$ in which the Bessel function $J_ν(x)$ is replaced by the modified Bessel functions $I_ν(x)$ and $K_ν(x)$ together with appropriate exponential factors $e^{\mp x}$, respectively.
The above integral with $J_ν(x)$ replaced by a hyper-Bessel function of the type recently discussed in Aktas {\it et al.} [The Ramanujan J., 2019] and taken over a finite interval determined by the first positive zero of the function is also considered for $n\to\infty$. We give the leading asymptotic behaviour of the hyper-Bessel function for $x\to+\infty$ in an appendix. Numerical examples are given to illustrate the accuracy of the various expansions obtained.
Discrete analogs of the index transforms, involving Bessel and the modified Bessel functions are introduced and investigated. The corresponding inversion theorems for suitable classes of functions and sequences are established.
Our aim is to extend some trigonometric inequalities to Bessel functions. Moreover, we extend the hyperbolic analogue of these trigonometric inequalities. As an application of these results we present a generalization of Cusa-type inequality to modified Bessel function. Our main motivation to write this paper is a recent publication of Chen and Sándor, which we wish to complement.
In the present paper, unification of Bessel, modified Bessel, spherical Bessel and Bessel-Clifford functions via the generalized Pochhammer symbol [ Srivastava HM, Cetinkaya A, Kıymaz O. A certain generalized Pochhammer symbol and its applications to hypergeometric functions. Applied Mathematics and Computation, 2014, 226 : 484-491] is defined. Several potentially useful properties of the unified family such as generating function, integral representation, Laplace transform and Mellin transform are obtained. Besides, the unified Bessel, modified Bessel, spherical Bessel and Bessel-Clifford fun…
In this paper our aim is to show some new inequalities of 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 use also 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 i…
In this note our aim is to present some monotonicity properties of the product of modified Bessel functions of first and second kind. Certain bounds for the product of modified Bessel functions of first and second kind are also obtained. These bounds improve and extend known bounds for the product of modified Bessel functions of 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.
The article considers the generalized k-Bessel functions and represents it as Wright functions. Then we study the monotonicity properties of the ratio of two different orders k- Bessel functions, and the ratio of the k-Bessel and the m-Bessel functions. The log-convexity with respect to the order of the k-Bessel also given. An investigation regarding the monotonicity of the ratio of the k-Bessel and k-confluent hypergeometric functions are discussed.
In this paper we give some conditions for a class of functions related to Bessel functions to be positive definite or strictly positive definite . We present some properties and relationships involving logarithmically completely monotonic functions and strictly positive definite functions. In particular, we are interested with the modified Bessel functions.
Simple inequalities are established for some integrals involving the modified Bessel functions of the first and second kind. In most cases these inequalities are tight in certain limits. As a consequence, we deduce a tight double inequality, involving the modified Bessel function of the first kind, for a generalized hypergeometric function. We also present some open problems that arise from this research.
Generalized integral formulas involving the generalized modified k-Bessel function $J_{k,ν}^{c,γ,λ}\left( z\right) $ of first kind are expressed in terms generalized $k-$Wright functions. Some interesting special cases of the main results are also discussed
In the paper, the author establishes inequalities, monotonicity, convexity, and unimodality for functions concerning the modified Bessel functions of the first kind and compute the completely monotonic degrees of differences between the exponential and trigamma functions.
In this paper, we obtain uniform bounds for a number of expressions that involve derivatives and integrals of modified Bessel functions. These uniform bounds are motivated by the need to bound such expressions in the study of variance-gamma and product normal approximations via Stein's method.
Simple inequalities for some integrals involving the modified Bessel functions $I_ν(x)$ and $K_ν(x)$ are established. We also obtain a monotonicity result for $K_ν(x)$ and a new lower bound, that involves gamma functions, for $K_0(x)$.
We obtain new inequalities for the modified Bessel function of the second kind $K_ν$ in terms of the gamma function. These bounds follow as special cases of inequalities that we derive for the kernel of the Krätzel integral transformation.
In this paper some new series and integral representations for the Turánian of modified Bessel functions of the first kind are given, which give new asymptotic expansions and tight bounds for the Turán determinant in the question. It is shown that in the case of natural and real order the Turánian can be represented in a relatively compact form, which yields a uniform upper bound for the Turán determinant for modified Bessel functions of the first kind. Our results complement and improve some of the results from the literature.
Motivated by some applications in applied mathematics, biology, chemistry, physics and engineering sciences, new tight Turán type inequalities for modified Bessel functions of the first and second kind are deduced. These inequalities provide sharp lower and upper bounds for the Turánian of modified Bessel functions of the first and second kind, and in most cases the relative errors of the bounds tend to zero as the argument tends to infinity. The chief tools in our proofs are some ideas of Gronwall [19] on ordinary differential equations, an integral representation of Ismail [28,29] for the qu…
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 kind, 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 …
We investigate the integral \[\int_0^\infty \cosh^μ\!t\,K_ν(z\cosh t)\,dt \qquad \Re(z)>0,\] where $K$ denotes the modified Bessel function, for non-negative integer values of the parameters $μ$ and $ν$. When the integers are of different parity, closed-form expressions are obtained in terms of $z^{-1}e^{-z}$ multiplied by a polynomial in $z^{-1}$ of degree dependent on the sign of $μ-ν$.
In this paper our aim is to show some mean value inequalities for the modified Bessel functions of the first and second kinds. Our proofs are based on some bounds for the logarithmic derivatives of these functions, which are in fact equivalent to the corresponding Turán type inequalities for these functions. As an application of the results concerning the modified Bessel function of the second kind we prove that the cumulative distribution function of the gamma-gamma distribution is log-concave. At the end of this paper several open problems are posed, which may be of interest for further rese…
Simple inequalities are established for some integrals involving the modified Bessel functions of the first and second kind. In most cases, we show that we obtain the best possible constant or that our bounds are tight in certain limits. We apply these inequalities to obtain uniform bounds for several expressions involving integrals of modified Bessel functions. Such expressions occur in Stein's method for variance-gamma approximation, and the results obtained in this paper allow for technical advances in the method. We also present some open problems that arise from this research.
The reality of the zeros of the product and cross-product of Bessel and modified Bessel functions of the first kind is studied. As a consequence the reality of the zeros of two hypergeometric polynomials is obtained together with the number of the Fourier critical points of the normalized forms of the product and cross-product of Bessel functions. Moreover, the interlacing properties of the real zeros of these products of Bessel functions and their derivatives are also obtained. As an application some geometric properties of the normalized forms of the cross-product and product of Bessel and m…
We employ the exponentially improved asymptotic expansions of the confluent hypergeometric functions on the Stokes lines discussed by the author [Appl. Math. Sci. {\bf 7} (2013) 6601--6609] to give the analogous expansions of the modified Bessel functions $I_ν(z)$ and $K_ν(z)$ for large $z$ and finite $ν$ on $\arg\,z=\pmπ$ (and, in the case of $I_ν(z)$, also on $\arg\,z=0$). Numerical results are presented to illustrate the accuracy of these expansions.
Recently Dixit, Kesarwani, and Moll introduced a generalization $K_{z,w}(x)$ of the modified Bessel function $K_{z}(x)$ and showed that it satisfies an elegant theory similar to $K_{z}(x)$. In this paper, we show that while $K_{\frac{1}{2}}(x)$ is an elementary function, $K_{\frac{1}{2},w}(x)$ can be written in the form of an infinite series of Humbert functions. As an application of this result, we generalize the transformation formula for the logarithm of the Dedekind eta function $η(z)$.
Fractional kinetic equations are investigated in order to describe the various phenomena governed by anomalous reaction in dynamical systems with chaotic motion. Many authors have provided solutions for various families of fractional kinetic equations involving special functions. Here, in this paper, we aim at presenting solutions of certain general families of fractional kinetic equations associated with the generalized modified $k$-Bessel function of the first kind. It is also pointed out that the main results presented here are general enough to be able to be specialized to yield many known…
The modified Bessel function of the second kind $K_{iν}(x)$ of imaginary order for fixed $x>0$ possesses a countably infinite sequence of real zeros. Recently it has been shown that the $n$th zero behaves like $ν_n\sim πn/\log\,n$ as $n\to\infty$. In this note we determine a more precise estimate for the bahaviour of these zeros for large $n$ by making use of the known asymptotic expansion of $K_{iν}(x)$ for large $ν$. Numerical results are presented to illustrate the accuracy of the expansion obtained.
Expressions for the derivatives with respect to order of modified Bessel functions evaluated at integer orders and certain integral representations of associated Legendre functions with modulus argument greater than unity are used to compute derivatives of the associated Legendre functions with respect to their parameters. For the associated Legendre functions of the first and second kind, derivatives with respect to the degree are evaluated at odd-half-integer degrees, for general complex orders, and derivatives with respect to the order are evaluated at integer orders, for general complex de…
New index transforms of the Lebedev type are investigated. It involves the real part of the product of the modified Bessel functions as the kernel. The boundedness and invertibility are examined for these operators in the Lebesgue weighted spaces. Inversion theorems are proved. Important particular cases are exhibited. The results are applied to solve an initial value problem for the fourth order PDE, involving the Laplacian. Finally, it is shown that the same PDE has another fundamental solution, which is associated with the generalized Lebedev index transform, involving the square of the mod…
The bounds for the ratios of first and second kind modified Bessel functions of consecutive orders are important quantities appearing in a large number of scientific applications. We obtain new bounds which are accurate in a large region of parameters and which are shaper than previous bounds. The new bounds are obtained by a qualitative analysis of the Riccati equation satisfied by these ratios. A procedure is considered in which the bounds obtained from the analysis of the Riccati equation are used to define a new function satisfying a new Riccati equation which yields sharper bounds. Simila…
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 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 a conjecture…
We examine the sum of modified Bessel functions with argument depending quadratically on the summation index given by \[S_ν(a)=\sum_{n\geq 1} (\frac{1}{2} an^2)^{-ν} K_ν(an^2)\qquad (|\arg\,a|<π/2)\] as the parameter $|a|\to 0$. It is shown that the positive real $a$-axis is a Stokes line, where an infinite number of increasingly subdominant exponentially small terms present in the asymptotic expansion undergo a smooth, but rapid, transition as this ray is crossed. Particular attention is devoted to the details of the expansion on the Stokes line as $a\to 0$ through positive values. Numerical …
A recent paper by C.G. Kokologiannaki published in J. Math. Anal. Appl. \cite{Kolo:2012:BFI} gives some properties for ratios of modified Bessel functions and, in particular, some bounds. These bounds are said to improve the range of some inequalities in \cite{Segura:2011:BRM} or to be sharper. Unfortunately, Kokologiannaki made some mistakes in the comparison and no improvement or extension is made over the results in \cite{Segura:2011:BRM}. We explain the errors in these comments and show that the bounds given in \cite{Kolo:2012:BFI} are already contained in \cite{Segura:2011:BRM}.
In this note we apply a modified fractional Bessel differential equation to the problem of describing corneal topography. We find the solution in terms of the power series. This solution has an interesting behavior at infinity which is a generalization of the classical results for modified Bessel function of order 0. Our model fits the real corneal geometry data with an error of order of a few per cent.
We examine the sum of modified Bessel functions with argument depending non-linearly on the summation index given by \[S_{ν,p}(a)=\sum_{n\geq 1} (an^p/2)^{-ν} K_ν(an^p)\qquad (a>0,\ 0\leqν<1)\] as the parameter $a\to 0+$, where $p$ denotes an integer satisfying $p\geq 2$. This extends previous work for the cases $p=1$ (linear) and $p=2$ (quadratic). The expansion as $a\to0+$ consists of an infinite number of asymptotic sums involving the Riemann zeta function, which when optimally truncated lead to remainder terms that are exponentially small in the parameter $a$. The number of these exponenti…
In this paper we study the Dini functions and the cross-product of Bessel functions. Moreover, we are interested on the monotonicity patterns for the cross-product of Bessel and modified Bessel functions. In addition, we deduce Redheffer-type inequalities, and the interlacing property of the zeros of Dini functions and the cross-product of Bessel and modified Bessel functions. Bounds for logarithmic derivatives of these functions are also derived. The key tools in our proofs are some recently developed infinite product representations for Dini functions and cross-product of Bessel functions.
The modified Bessel functions $K_ν(z)$, or, for brevity, K-Bessel functions, arise at key places in analytic number theory. In particular, they appear in beautiful arithmetic identities. A survey of these arithmetical identities and their appearances in number theory is provided.
We have used recent integral representations of the derivatives of the Bessel functions with respect to the order to obtain closed-form expressions in terms of generalized hypergeometric functions and Meijer-$G$ functions. Also, we have carried out similar calculations for the derivatives of the modified Bessel functions with respect to the order, obtaining closed-form expressions as well. For this purpose, we have obtained integral representations of the derivatives of the modified Bessel functions with respect to the order. As by-products, we have calculated two non-tabulated integrals.
We examine convergent representations for the sum of Bessel functions \[\sum_{n=1}^\infty \frac{J_μ(na) J_ν(nb)}{n^α}\] for $μ$, $ν\geq0$ and positive values of $a$ and $b$. Such representations enable easy computation of the series in the limit $a, b\to0+$. Particular attention is given to logarithmic cases that occur both when $a=b$ and $a\neq b$ for certain values of $α$, $μ$ and $ν$. The series when the first Bessel function is replaced by the modified Bessel function $K_μ(na)$ is also investigated, as well as the series with two modified Bessel functions.
The $ν$-zeros of the Bessel functions of purely imaginary order are examined for fixed argument $x>0$. In the case of the modified Bessel function of the second kind $K_{iν}(x)$, it is known that it possesses a countably infinite sequence of real $ν$-zeros described by $ν_n\sim πn/\log\,n$ as $n\to\infty$. Here we apply a unified approach to determine asymptotic estimates of the $ν$-zeros of the modified Bessel functions $L_{iν}(x)\equiv I_{iν}(x)+I_{-iν}(x)$ and $K_{iν}(x)$ and the ordinary Bessel functions $J_{iν}(x)\pm J_{-iν}(x)$.
Simple upper and lower bounds are obtained for the integral $\int_0^x\mathrm{e}^{-γt}t^νI_ν(t)\,\mathrm{d}t$, $x>0$, $ν>-\frac{1}{2}$, $0<γ<1$. Most of our bounds for this integral are tight as $x\rightarrow\infty$. We apply one of our inequalities to bound some expressions involving this integral. Two of these expressions appear in Stein's method for variance-gamma approximation, and our bounds will allow for a technical advancement to be made to the method.
In this paper, we are interested in a Neumann-type series for modified Bessel functions of the first kind which arises in the study of Dunkl operators associated with dihedral groups and as an instance of the Laguerre semigroup constructed by Ben Said-Kobayashi-Orsted. We first revisit the particular case corresponding to the group of square-preserving symmetries for which we give two new and different proofs other than the existing ones. The first proof uses the expansion of powers in a Neumann series of Bessel functions while the second one is based on a quadratic transformation for the Gaus…
In this paper we consider a sum of modified Bessel functions of the first kind of which particular case is used in the study of Kanter's sharp modified Bessel function bound for concentrations of some sums of independent symmetric random vectors. We present some monotonicity and convexity properties for that sum of modified Bessel functions of the first kind, as well as some Turán type inequalities, lower and upper bounds. Moreover, we point out an error in Kanter's paper [Ka] and at the end of the paper we pose an open problem, which may be of interest for further research.
We deduce some new functional inequalities, like Turán type inequalities, Redheffer type inequalities, and a Mittag-Leffler expansion for a special combination of modified Bessel functions of the first kind, called modified Dini functions. Moreover, we show the complete monotonicity of a quotient of modified Dini functions by introducing a new continuous infinitely divisible probability distribution. The key tool in our proofs is a recently developed infinite product representation for a special combination of Bessel functions of the first, which was very useful in determining the radius of co…
In this paper a necessary and sufficient condition is deduced for the close-to-convexity of a cross product of Bessel and modified Bessel functions of the first kind and their derivatives by using a result of Shah and Trimble about transcendental entire functions with univalent derivatives, the newly discovered power series and infinite product representation of this cross-product, as well as a slightly modified version of a result of Lorch on the monotonicity of the zeros of the cross product with respect to the order.
The goal of this paper is to extend the classical and multiplicative fractional derivatives. For this purpose, it is introduced the new extended modified Bessel function and also given an important relation between this new function I(v,q;x) and the confluent hypergeometric function. Besides, it is used to generalize the hypergeometric, the confluent hypergeometric and the extended beta functions by using the new extended modified Bessel function. Also, the asymptotic formulae and the generating function of the extended modified Bessel function are obtained. The extensions of classical and mul…
Let $K_{0}$ denote the modified Bessel function of second kind and zeroth order. In this paper we will studying the function $\tildeω_{n}\left( x\right) :=\frac{\left( -x\right) ^{n}K_{0}^{\left( n\right) }\left( x\right) }{n!}$ for positive argument. The function $\tildeω_{n}$ plays an important role for the formulation of the wave equation in two spatial dimensions as a retarded potential integral equation. We will prove that the growth of the derivatives $\tildeω_{n}^{\left( m\right) }$ with respect to $n$ can be bounded by $O\left( \left( n+1\right) ^{m/2}\right) $ while for small and larg…
We consider the asymptotic expansion of the Humbert hyper-Bessel function expressed in terms of a ${}_0F_2$ hypergeometric function by \[J_{m,n}(x)=\frac{(x/3)^{m+n}}{m! n!}\,{}_0F_2(-\!\!\!-;m+1, n+1; -(x/3)^3)\] as $x\to+\infty$, where $m$, $n$ are not necessarily non-negative integers. Particular attention is paid to the determination of the exponentially small contribution. The main approach utilised is that described by the author (J. Comput. Appl. Math. {\bf 234} (2010) 488-504); a leading-order estimate is also obtained by application of the saddle-point method applied to an integral re…
Let $I_ν(x)$ and $K_ν(x)$ be the first and second kind modified Bessel functions. It is shown that the nullclines of the Riccati equation satisfied by $x^α Φ_{i,ν}(x)$, $i=1,2$, with $Φ_{1,ν}=I_{ν-1}(x)/I_ν(x)$ and $Φ_{2,ν}(x)=-K_{ν-1}(x)/K_ν(x)$, are bounds for $x^α Φ_{i,ν}(x)$, which are solutions with unique monotonicity properties; these bounds hold at least for $\pm α\notin (0,1)$ and $ν\ge 1/2$. Properties for the product $P_ν(x)=I_ν(x)K_ν(x)$ can be obtained as a consequence; for instance, it is shown that $P_ν(x)$ is decreasing if $ν\ge -1$ (extending the known range of this result) an…
In the paper, the authors verify the complete monotonicity of the difference $e^{1/t}-ψ'(t)$ on $(0,\infty)$, compute the completely monotonic degree and establish integral representations of the remainder of the Laurent series expansion of $e^{1/z}$, and derive an inequality which gives a lower bound for the first order modified Bessel function of the first kind.
In this present paper our aim is to deal with two integral transforms which involving the Gauss hypergeometric function as its kernels. We prove some compositions formulas for such a generalized fractional integrals with k Bessel function. The results are established in terms of generalized Wright type hypergeometric function and generalized hypergeometric series. Also, the authors presented some corresponding assertions for Riemann Liouville and Erdelyi Kober fractional integral transforms.
Linear second order differential equations of the form $d^{2}w/dz^{2}-\left \{ {u^{2}f\left( u,z\right) +g\left( z\right) }\right\} w=0$ are studied, where $\left| u\right| \rightarrow \infty $ and $z$ lies in a complex bounded or unbounded domain $\mathbf{D}$. If $f\left( u,z\right) $ and $g\left( z\right) $ are meromorphic in $\mathbf{D}$, and $f\left(u,z\right) $ has no zeros, the classical Liouville-Green/WKBJ approximation provides asymptotic expansions involving the exponential function. The coefficients in these expansions either multiply the exponential, or in an alternative form appea…
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 further obtain a convergent expansion of the generalized hypergeometric function ${}_{p}F_p$ in terms of the confluent hypergeometric functions ${}_{1}F_1$ that holds uniformly in any right half-plane. For both functions, we make a further step and give convergent expansions in terms of trigonometric, exponential and rational functions that hold uniformly in the s…
Discrete analogs of the Lebedev transforms with the product of the modified Bessel functions are introduced and investigated. Several expansions of suitable functions and sequences in terms of the series and integrals, involving the modified and incomplete Bessel functions are established.
The order derivatives of the modified Bessel function of the second kind at s = .5 are obtained as finite expressions of integrals that generalize the exponential integral appearing in the first derivative (Theorem 1.) The derivatives arise in the investigation of a BesselK relationship with the Riemann zeta function. Any use of the term, derivative, with respect to a Bessel function, here means a derivative with respect to its order.
New index transforms are investigated, which contain as the kernel products of the Bessel and modified Bessel functions. Mapping properties and invertibility in Lebesgue spaces are studied for these operators. Relationships with the Kontorovich-Lebedev and Fourier cosine transforms are established. Inversion theorems are proved. As an application, a solution of the initial value problem for the fourth order partial differential equation, involving the Laplacian is presented.
We consider two sequences $a(n)$ and $b(n)$, $1\leq n<\infty$, generated by Dirichlet series $$\sum_{n=1}^{\infty}\frac{a(n)}{λ_n^{s}}\qquad\text{and}\qquad \sum_{n=1}^{\infty}\frac{b(n)}{μ_n^{s}},$$ satisfying a familiar functional equation involving the gamma function $Γ(s)$. Two general identities are established. The first involves the modified Bessel function $K_μ(z)$, and can be thought of as a 'modular' or 'theta' relation wherein modified Bessel functions, instead of exponential functions, appear. Appearing in the second identity are $K_μ(z)$, the Bessel functions of imaginary argument…
