Kummer Functions Research Articles

On some fractional generalizations of the Laguerre polynomials and the Kummer function

This paper refers to some generalizations of the classical Laguerre polynomials. By means of the Riemann–Liouville operator of fractional calculus and Rodrigues’ type representation formula of fractional order, the Laguerre functions are derived and some of their properties are given and compared with the corresponding properties of the classical Laguerre polynomials. Further generalizations of the Laguerre functions are introduced as a solution of a fractional version of the classical Laguerre differential equation. Likewise, a generalization of the Kummer function is introduced as a solution of a fractional version of the Kummer differential equation. The Laguerre polynomials and functions are presented as special cases of the generalized Laguerre and Kummer functions. The relation between the Laguerre polynomials and the Kummer function is extended to their fractional counterparts.

Patterns of high energy massive string scatterings in the Regge regime

We calculate high energy massive string scattering amplitudes of open bosonic string in the Regge regime (RR). We found that the number of high energy amplitudes for each fixed mass level in the RR is much more numerous than that of Gross regime (GR) calculated previously. Moreover, we discover that the leading order amplitudes in the RR can be expressed in terms of the Kummer function of the second kind. In particular, based on a summation algorithm for Stirling number identities developed recently, we discover that the ratios calculated previously among scattering amplitudes in the GR can be extracted from this Kummer function in the RR. We conjecture and give evidences that the existence of these GR ratios in the RR persists to subleading orders in the Regge expansion of all string scattering amplitudes. Finally, we demonstrate the universal power-law behavior for all massive string scattering amplitudes in the RR.

Analytical solution of Graetz problem in pipe flow with periodic inlet temperature variation

The paper reports an analytical solution for the temperature field in a fully developed pipe flow subject to periodic (of any shape) inlet temperature variation. The solution is given in term of a series of Kummer functions for the cases of uniform and constant wall temperature and wall heat flux, thus comprising also the adiabatic wall case. A “fully developed” region for the fluctuating component of the fluid temperature is also evidenced and closed-form solutions are given. An interpretation of the temperature field as superposition of travelling thermal waves is presented and discussed.

Computational properties of three-term recurrence relations for Kummer functions

Several three-term recurrence relations for confluent hypergeometric functions are analyzed from a numerical point of view. Minimal and dominant solutions for complex values of the variable z are given, derived from asymptotic estimates of the Whittaker functions with large parameters. The Laguerre polynomials and the regular Coulomb wave functions are studied as particular cases, with numerical examples of their computation.

Nonparaxial hypergeometric beams

We derive an analytical expression to describe the exact solution of the Helmholtz equation in cylindrical coordinates as a product of two Kummer functions. The solution is presented as a sum of two terms that describe the nonparaxial hypergeometric light beams propagated along the optical axis in the positive and negative directions. With the distance from the initial plane becoming much larger than the wavelength of light, the expression derived for the nonparaxial hypergeometric beam coincides with that for a paraxial hypergeometric mode.

Three-dimensional optical vortex and necklace solitons in highly nonlocal nonlinear media

We demonstrate the existence of localized optical vortex and necklace solitons in three-dimensional (3D) highly nonlocal nonlinear media, both analytically and numerically. The 3D solitons are constructed with the help of Kummer's functions in spherical coordinates and their unique properties are discussed. The procedure we follow offers ways for generation, control, and manipulation of spatial solitons.

Kummer solitons in strongly nonlocal nonlinear media

We solve the three-dimensional (3D) time-dependent strongly nonlocal nonlinear Schrödinger equation (NNSE) in spherical coordinates, with the help of Kummer's functions. We obtain analytical solitary solutions, which we term the Kummer solitons. We compare analytical solutions with the numerical solutions of NNSE. We discuss higher-order Kummer spatial solitons, which can exist in various forms, such as the 3D vortex solitons and the multipole solitons.

Fresnel and Fraunhofer diffraction of a Gaussian laser beam by fork-shaped gratings

Expressions describing the vortex beams that are generated by the process of Fresnel diffraction of a Gaussian beam incident out of waist on fork-shaped gratings of arbitrary integer charge p, and vortex spots in the case of Fraunhofer diffraction by these gratings, are deduced. The common general transmission function of the gratings is defined and specialized for the cases of amplitude holograms, binary amplitude gratings, and their phase versions. Optical vortex beams, or carriers of phase singularity with charges mp and -mp, are the higher negative and positive diffraction-order beams. The radial part of their wave amplitudes is described by the product of the mpth-order Gauss-doughnut function and a Kummer function, or by the first-order Gauss-doughnut function and the difference of two modified Bessel functions whose orders do not match the singularity charge value. The wave amplitude and the intensity distributions are discussed for the near and far fields in the focal plane of a convergent lens, as well as the specialization of the results when the grating charge p=0; i.e., the grating turns from forked into rectilinear. The analytical expressions for the vortex radii are also discussed.

A note on Turán type and mean inequalities for the Kummer function

Turán-type inequalities for combinations of Kummer functions involving Φ ( a ± ν , c ± ν , x ) and Φ ( a , c ± ν , x ) have been recently investigated in [Á. Baricz, Functional inequalities involving Bessel and modified Bessel functions of the first kind, Expo. Math. 26 (3) (2008) 279–293; M.E.H. Ismail, A. Laforgia, Monotonicity properties of determinants of special functions, Constr. Approx. 26 (2007) 1–9]. In the current paper, we resolve the corresponding Turán-type and closely related mean inequalities for the additional case involving Φ ( a ± ν , c , x ) . The application to modeling credit risk is also summarized.

HEAT TRANSFER IN MHD VISCOELASTIC FLUID FLOW ON STRETCHING SHEET WITH HEAT SOURCE/SINK, VISCOUS DISSIPATION, STRESS WORK, AND RADIATION FOR THE CASE OF LARGE PRANDTL NUMBER

An analysis is carried out to study the magnetohydrodynamic boundary layer flow behavior and heat transfer characteristics of a viscoelastic fluid flow over a stretching sheet with radiation and for the case of large Prandtl numbers. The basic boundary layer equations of momentum and heat transfer, which are nonlinear partial differential equations, are converted into nonlinear ordinary differential equations by means of similarity transformation. The resulting nonlinear ordinary differential equations of momentum are solved exactly. Similarly, the energy equation is transformed to a confluent hypergeometric differential equation using a new variable and Rosseland approximation for radiation. The analytic solutions for temperature profile and heat transfer characteristics are obtained in terms of Kummer's function, and their asymptotic limits for large Prandtl numbers are also obtained. The effects of magnetic field, viscoelastic parameter, viscous dissipation, heat generation/absorption, work done due to deformation, and radiation on flow and heat transfer characteristics are discussed through several graphs. To assess the validity and accuracy of the present work, heat transfer results were compared to those of previously published work of Nataraja et al. (1977). This comparison shows excellent agreement between the results.

Derivatives of any order of the confluent hypergeometric function F11(a,b,z) with respect to the parameter a or b

The derivatives to any order of the confluent hypergeometric (Kummer) function F=F11(a,b,z) with respect to the parameter a or b are investigated and expressed in terms of generalizations of multivariable Kampé de Fériet functions. Various properties (reduction formulas, recurrence relations, particular cases, and series and integral representations) of the defined hypergeometric functions are given. Finally, an application to the two-body Coulomb problem is presented: the derivatives of F with respect to a are used to write the scattering wave function as a power series of the Sommerfeld parameter.

Heat source and radiation effects on magneto-convection flow of a viscoelastic fluid past a stretching sheet: Analysis with Kummer's functions

An analysis is carried out to study heat source and radiation effects on two-dimensional steady flow of an electrically conducting, incompressible, viscoelastic fluid ( Walter's liquid-B′) past a stretching sheet in the presence of transverse uniform magnetic field. Two cases are studied namely (i) the sheet with prescribed power law surface temperature (PST case) and (ii) the sheet with prescribed power law surface heat flux (PHF case). Kummer's functions are used to obtain temperature field and wall temperature gradient. The variations in the velocity and temperature field with change in parameters encountered into the equations are obtained and depicted graphically. The numerical values of the variations in wall temperature gradient due to change in physical parameters are presented in the tables. The results obtained have been discussed.

Expressions for angular mean and variance

Expressions are derived for the mean and variance of the large sample approximation of the type II and IV angles estimators considered by Schober [1]. The expressions involve the well-known Kummer function.

Kummer function solutions of damped Burgers equations with time-dependent viscosity by exact linearization

Through a Cole-Hopf like transformation a generalized Burgers equation is linearized to Kummer's equation. The large time behaviour of solutions of the GBE is determined. Some exact closed form solutions are also found.

Asymptotics of the zeros of degenerate hypergeometric functions

We find the asymptotics of the zeros of the degenerate hypergeometric function (the Kummer function) Φ(a, c; z) and indicate a method for numbering all of its zeros consistent with the asymptotics. This is done for the whole class of parameters a and c such that the set of zeros is infinite. As a corollary, we obtain the class of sine-type functions with unfamiliar asymptotics of their zeros. Also we prove a number of nonasymptotic properties of the zeros of the function Φ.

Market switching in shipping — A real option model applied to the valuation of combination carriers

This paper derives a real options model of flexibility and applies it to shipping, valuing the option to switch between the dry bulk market and wet bulk market for a combination carrier, a ship type that is capable of operating in both markets but that has fallen out of favor due to high price tags. The model is a mean-reverting (Ornstein–Uhlenbeck) version of a standard entry–exit model with stochastic prices. A closed form solution for the value of flexibility is derived, expressed in terms of Kummer functions. The estimated value of flexibility is related to historical price differentials between combination carriers and oil tankers of comparable size. Based on numerical experiments it is concluded that new combination carriers may enter the market in the near future.

Some multivariable Gaussian hypergeometric extensions of the Preece theorem

AbstractSome generalisations of the Preece theorem involving the product of two Kummer's functions 1F1 are obtained using Dixon's theorem and some well-known identities. Its special cases yield various new transformations and reduction formulae involving Pathan's quadruple hypergeometric function and Srivastava's quadruple hypergeometric function F(4) and triple hypergeometric function F(3). Some known results of Preece, Pathan and Bailey are also obtained as special cases.

Functional inequalities involving special functions II

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.

A diffusion model with cubic drift: statistical and computational aspects and application to modelling of the global CO2 emission in Spain

AbstractThe aim of this work is the study of a new stochastic diffusion model with a cubic‐type drift coefficient. The model is considered as the solution of an Ito stochastic differential equation. Using the Ito's stochastic calculus and properties of the Kummer function, the trend functions and steady‐state distribution for the process are obtained. Statistical estimation and corresponding computational methodology are established. Finally, the model is applied to modelling and prediction of the global CO2 emission in Spain. Copyright © 2006 John Wiley & Sons, Ltd.

Analytical solution of multimodal acoustic propagation in circular ducts with laminar mean flow profile

In this paper, an analytical solution for the propagation of sound in circular ducts in the presence of Poiseuille mean flow is derived. This solution uses Kummer's functions and it generalizes the results found by Gogate and Munjal. Links between this solution and already-known solutions for more specific cases (no mean flow, uniform flow) are made. Results for calculation of the propagation constant and pressure profiles are presented and compared with the literature. The effect of shear on multimodal acoustic propagation, including modes of higher radial order is discussed.