Lommel Functions Research Articles

Lommel functions, Padé approximants and hypergeometric functions

We consider the Lommel functions sμ,ν(z) for different values of the parameters (μ,ν). We show that if (μ,ν) are half integers, then it is possible to describe these functions with an explicit combination of polynomials and trigonometric functions. The polynomials turn out to give Padé approximants for the trigonometric functions. Numerical properties of the zeros of the polynomials are discussed. Also, when μ is an integer, sμ,ν(z) can be written as an integral involving an explicit combination of trigonometric functions. A closed formula for F12(12+ν,12−ν;μ+12;sin⁡(θ2)2) with μ an integer is given.

Absolute moments of the variance-gamma distribution

We obtain exact formulas for the absolute raw and central moments of the variance-gamma distribution, as infinite series involving the modified Bessel function of the second kind and the modified Lommel function of the first kind. When the skewness parameter is equal to zero (the symmetric variance-gamma distribution), the infinite series reduces to a single term. Moreover, for the case that the shape parameter is a half-integer (in our parameterisation of the variance-gamma distribution), we obtain a closed-form expression for the absolute moments in terms of confluent hypergeometric functions. As a consequence, we deduce new exact formulas for the absolute raw and central moments of the asymmetric Laplace distribution and the product of two correlated zero mean normal random variables, and more generally the sum of independent copies of such random variables.

Integral Representations and Zeros of the Lommel Function and the Hypergeometric 1F2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$_1F_2$$\\end{document} Function

We give different integral representations of the Lommel function sμ,ν(z)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$s_{\\mu ,\ u }(z)$$\\end{document} involving trigonometric and hypergeometric 2F1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$_2F_1$$\\end{document} functions. By using classical results of Pólya, we give the distribution of the zeros of sμ,ν(z)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$s_{\\mu ,\ u }(z)$$\\end{document} for certain regions in the plane (μ,ν)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\mu ,\ u )$$\\end{document}. Further, thanks to a well known relation between the functions sμ,ν(z)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$s_{\\mu ,\ u }(z)$$\\end{document} and the hypergeometric 1F2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ _1F_2$$\\end{document} function, we describe the distribution of the zeros of 1F2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$_1F_2$$\\end{document} for specific values of its parameters.

Monotonicity patterns and functional inequalities for modified Lommel functions of the first kind

Monotonicity patterns and functional inequalities for modified Lommel functions of the first kind

ON THE CUMULATIVE DISTRIBUTION FUNCTION OF THE VARIANCE-GAMMA DISTRIBUTION

AbstractWe obtain exact formulas for the cumulative distribution function of the variance-gamma distribution, as infinite series involving the modified Bessel function of the second kind and the modified Lommel function of the first kind. From these formulas, we deduce exact formulas for the cumulative distribution function of the product of two correlated zero-mean normal random variables.

The Monotony of the Lommel Functions

The Monotony of the Lommel Functions

On the characterization properties of certain hypergeometric functions in the open unit disk

Our purpose in the present investigation is to study certain geometric properties such as the close-to-convexity, convexity, and starlikeness of the hypergeometric function z{}_{1}F_{2}(a;b,c;z) in the open unit disk. The usefulness of the main results for some familiar special functions like the modified Sturve function, the modified Lommel function, the modified Bessel function, and the {}_{0}F_{1}(-;c;z) function are also mentioned. We further consider a boundedness property of the function _{1}F_{2}(a;b,c;z) in the Hardy space of analytic functions. Several corollaries and special cases of the main results are also pointed out.

Characterizations for Certain Subclasses of Starlike and Convex Functions Associated with Lommel, Struve and Bessel Functions

In this paper we have determined some necessary and sufficient conditions of normalized Lommel function sμ,v, Struve function hv and Bessel function jv of the first kind to be in the subclasses S(α, β, γ) and K(α, β, γ) of starlike and convex functions of order α and type β, γ, in the unit disk U.

A Triple Integral Containing the Lommel Function su,v(z): Derivation and Evaluation

A three-dimensional integral containing the kernel g(x, y, z)su,v(z) is derived. The function g(x, y, z) is a generalized function containing the logarithmic and exponential functions and su,v(z) is the Lommel function and the integral is taken over the cube 0 ≤ y ≤ ∞, 0 ≤ x ≤∞, 0 ≤ z ≤ ∞. A representation in terms of the Lerch function is derived, from which special cases can be evaluated. Almost all Hurwitz-Lerch Zeta functions have an asymmetrical zero distribution. All the results in this work are new.

Bounds for an Integral Involving the Modified Lommel Function of the First Kind

Simple upper and lower bounds are established for the integral int _0^xmathrm {e}^{-beta u}u^nu t_{mu ,nu }(u),mathrm {d}u, where x>0, 0<beta <1, mu +nu >-2, mu -nu ge -3, and t_{mu ,nu }(x) is the modified Lommel function of the first kind. Our bounds complement and improve on existing bounds for this integral, by either being sharper or increasing the range of validity. Our bounds also generalise recent bounds for an integral involving the modified Struve function of the first kind, and in some cases more a direct approach lead to sharper bounds when our general bounds are specialised to the modified Struve case.

Radius of k-Parabolic Starlikeness for Some Entire Functions

This article considers three types of analytic functions based on their infinite product representation. The radius of the k-parabolic starlikeness of the functions of these classes is studied. The optimal parameter values for k-parabolic starlike functions are determined in the unit disk. Several examples are provided that include special functions such as Bessel, Struve, Lommel, and q-Bessel functions.

Analytical Solutions to Definite Integrals for Combinations of Legendre, Bessel and Trigonometric Functions Encountered in Propagation and Scattering Problems in Spherical Coordinates

Mie theory is a rigorous solution to scattering problems in spherical coordinate system. The first step in applying Mie theory is expansion of some arbitrary incident field in terms of spherical harmonics fields in terms of spherical which in turn requires evaluation of certain definite integrals whose integrands are products of Bessel functions, associated Legendre functions and periodic functions. Here we present analytical results for two specific integrals that are encountered in expansion of arbitrary fields in terms of summation of spherical waves. The analytical results are in terms of finite summations which include Lommel functions. A concise analytical expression is also derived for the special case of Lommel functions that arise, rendering expensive numerical integration or other iterative techniques unnecessary.

Free Space Realization of the Symmetrical Tunable Auto‐Focusing Lommel Gaussian Vortex Beam

AbstractAn auto‐focusing beam with adjustable symmetry is proposed numerically and experimentally. The initial field of the auto‐focusing Lommel Gaussian vortex beam (ALoGVB) consists of the Lommel function and the Gaussian term. When the asymmetry factor ε of the beam approaches zero, the ALoGVB will degenerate into a circular Bessel Gaussian vortex beam (CBGVB). Compared with the CBGVB, the ALoGVB is more diverse. The shape, symmetry, normalized focusing intensity, and focus distance of the ALoGVB can be controlled by flexibly adjusting the asymmetry factor ε and the topological charge n. In addition, the effect of off‐axis vortices on the propagating characteristics of the beam is also discussed. The experiment is in good agreement with the numerical simulation. The results could be beneficial for potential applications in optical trapping or micromachining.

Functional Inequalities and Monotonicity Results for Modified Lommel Functions of the First Kind

We establish some monotonicity results and functional inequalities for modified Lommel functions of the first kind. In particular, we obtain new Turán type inequalities and bounds for ratios of modified Lommel functions of the first kind, as well as the function itself. These results complement and in some cases improve on existing results, and also generalise a number of the results from the literature on monotonicity patterns and functional inequalities for the modified Struve function of the first kind.

Uniform asymptotic expansions for Lommel, Anger‐Weber, and Struve functions

AbstractUsing a differential equation approach, asymptotic expansions are rigorously obtained for Lommel, Weber, Anger–Weber, and Struve functions, as well as Neumann polynomials, each of which is a solution of an inhomogeneous Bessel equation. The approximations involve Airy and Scorer functions, and are uniformly valid for large real order and unbounded complex argument . An interesting complication is the identification of the Lommel functions with the new asymptotic solutions, and in order to do so, it is necessary to consider certain sectors of the complex plane, as well as introduce new forms of Lommel and Struve functions.

Discrete index transformations with Bessel and Lommel functions

Discrete analogs of the index transforms, involving Bessel and Lommel functions are introduced and investigated. The corresponding inversion theorems for suitable classes of functions and sequences are established.

Radii of k-starlikeness of order α of Struve and Lommel functions

Radii of k-starlikeness of order α of Struve and Lommel functions

Univalence and convexity conditions for certain integral operators associated with the Lommel function of the first kind

&lt;abstract&gt;&lt;p&gt;A useful family of integral operators and special functions plays a crucial role on the study of mathematical and applied sciences. The purpose of the present paper is to give sufficient conditions for the families of integral operators, which involve the normalized forms of the generalized Lommel functions of the first kind to be univalent in the open unit disk. Furthermore, we determine the order of the convexity of the families of integral operators. In order to prove main results, we use differential inequalities for the Lommel functions of the first kind together with some known properties in connection with the integral operators which we have considered in this paper. We also indicate the connections of the results presented here with those in several earlier works on the subject of our investigation. Moreover, some graphical illustrations are provided in support of the results proved in this paper.&lt;/p&gt;&lt;/abstract&gt;

Exponential Starlikeness and Convexity of Confluent Hypergeometric, Lommel, and Struve Functions

Sufficient conditions are obtained on the parameters of Lommel function of the first kind, generalized Struve function of the first kind, and the confluent hypergeometric function under which these special functions become exponential convex and exponential starlike in the open unit disk. The method of differential subordination is employed in proving the results. Few examples are also provided to illustrate the results obtained.

Unified integral associated with the generalized V-function

In this paper, we present two new unified integral formulas involving a generalized V-function. Some interesting special cases of the main results are also considered in the form of corollaries. Due to the general nature of the V-function, several results involving different special functions such as the exponential function, the Mittag-Leffler function, the Lommel function, the Struve function, the Wright generalized Bessel function, the Bessel function and the generalized hypergeometric function are obtained by specializing the parameters in the presented formulas. More results are also discussed in detail.