Confluent Hypergeometric Functions Research Articles

Sufficient Conditions for Linear Operators Related to Confluent Hypergeometric Function and Generalized Bessel Function of the First Kind to Belong to a Certain Class of Analytic Functions

Geometric function theory has extensively explored the geometric characteristics of analytic functions within symmetric domains. This study analyzes the geometric properties of a specific class of analytic functions employing confluent hypergeometric functions and generalized Bessel functions of the first kind. Specific constraints are imposed on the parameters to ensure the inclusion of the confluent hypergeometric function within the analytic function class. The coefficient bound of the class is used to determine the inclusion properties of integral operators involving generalized Bessel functions of the first kind. Different results are observed for these operators, depending on the specific values of the parameters. The results presented here include some previously published findings as special cases.

A new generalized beta function associated with statistical distribution and fractional kinetic equation

Several authors have extensively investigated beta function, hypergeometric function and confluent hypergeometric function, their extensions and generalizations due to their several application in many areas of engineering, probability theory and science. The main purpose of this paper is to present a new generalization of extended beta function, hypergeometric function and confluent hypergeometric function with the help of m-parameter Mittag-Leffler function, as well as examine some important properties like integral representations, differential formula and summation formulas. We also examine generalized Caputo fractional derivative operator with the help of m-parameter Mittag-Leffler function with associated properties using the generalized beta function. We define a new beta distribution involving the new generalized beta function. The mean, variance, coefficient of variance, moment generating function and characteristic function and cumulative distribution are derived. Further, we derive the solution of fractional Kinetic equation involving generalized hypergeometric function.

An Analytical Study on the influence of Magnetic Field-Dependent Viscosity on Viscous and Ohmic Dissipative Second-Grade Fluid Boundary Layers

Abstract The article aims to investigate the magnetohydrodynamic (MHD) boundary layer flow of a second-grade non-Newtonian fluid over a horizontally stretching sheet, considering magnetic field-dependent (MFD) viscosity, as well as viscous and Ohmic dissipations. Analytical solutions for previously unexplored momentum and heat transfer equations involving MFD viscosity are derived. Two boundary conditions, namely Prescribed Surface Temperature (PST) and Prescribed Heat Flux (PHF), are taken into account. Governing dimensional partial differential equations (PDEs) are transformed into non-dimensional ordinary differential equations (ODEs) using similarity transformations. Closed-form analytical solutions for flow are derived, considering stretching velocity, MFD viscosity, second-grade fluid properties, and suction impacts. Heat transfer equations are transformed into Gauss-hypergeometric form, yielding solutions in terms of confluent hypergeometric functions. Analytical expressions for skin friction, local Nusselt number, and non-dimensional wall temperature are derived. A unique solution is obtained for the flow equation. It is found that both second-grade and magnetic viscosity parameters expand the momentum boundary layer, while the magnetic parameter reduces thickness. Thermal boundary layer thickness increases with a higher second-grade parameter, magnetic viscosity, and Eckert number. Moreover, analytical solutions are validated against published results for a special case.

Coulomb effect on elastic electron scattering by hydrogen atom and hydrogen-like ions in their metastable 2s state in the 511–770 KeV energy range

This paper reports on a study investigating the elastic scattering of hydrogen atoms in their metastable state (2S-2S) when subjected to electron impact, with a focus on the Coulomb effect within an energy range spanning from 511 keV to 770 keV . The analysis employs semi-relativistic wave functions to depict the hydrogen states. The validity of using Darwin wave functions is confirmed by ensuring that the condition Zα << 1 is met, where α represents the structure constant. Both incident and scattered free electrons are described using Dirac spinors. Notably, the semi-relativistic calculations converge with those of the non-relativistic case at low electron velocities. Additionally, in the presence of the Coulomb effect, the electron wave function is approximated by a Dirac function modulated by the confluent hypergeometric function. The paper presents the relativistic differential cross-section (RDCS) as a function of the final scattering angle for various energies of the incident electron, revealing intriguing insights into diffusion probabilities in specific directions. Moreover, the impact of the atomic number Z on the elastic scattering of hydrogen atoms and hydrogen-like ions by electron impact under the influence of the Coulomb effect is thoroughly examined.

Analytical Model of Point Spread Function under Defocused Degradation in Diffraction-Limited Systems: Confluent Hypergeometric Function

In recent years, optical systems near the diffraction limit have been widely used in high-end applications. Evidently, an analytical solution of the point spread function (PSF) will help to enhance both understanding and dealing with the imaging process. This paper analyzes the Fresnel diffraction of diffraction-limited optical systems in defocused conditions. For this work, an analytical solution of the defocused PSF was obtained using the series expansion of the confluent hypergeometric functions. The analytical expression of the defocused optical transfer function is also presented herein for comparison with the PSF. Additionally, some characteristic parameters for the PSF are provided, such as the equivalent bandwidth and the Strehl ratio. Comparing the PSF obtained using the fast Fourier transform algorithm of an optical system with known, detailed parameters to the analytical solution derived in this paper using only the typical parameters, the root mean square errors of the two methods were found to be less than 3% in the weak and medium defocus range. The attractive advantages of the universal model, which is independent of design details, objective types, and applications, are discussed.

Modified special functions: properties, integral transforms and applications to fractional differential equations

In this paper, we defined the modified gamma and beta functions, which involving the generalized M-series at their kernels, and then defined the modified Gauss and confluent hypergeometric functions by using the modified beta function. Furthermore, we presented some of their properties such that, integral representations, summation formulas and derivative formulas. Also, we applied beta, Mellin, Laplace, Sumudu, Elzaki and general integral transforms to newly defined the modified special functions. Then, we obtained solution of fractional differential equations involving the modified special functions, as examples. Finally, we gave the relationships between the modified functions with some of the generalized special functions, which can be found in the literature.

A NEW EXTENSION OF LOGARITHMIC BETA FUNCTION AND THEIR PROPERTIES

Khan Et Al., presented a new kind of beta logarithmic function, we aim in this research article to introduce new extension of beta logarithmic function, Further, we study its fundamental properties and discuss diverse formulas of that extension such as integral representation, summation formula, transform formula and their statistical properties. Based on this concept, we introduce new hypergeometric and confluent hypergeometric functions and study their properties.

Some Fourier transforms involving confluent hypergeometric functions

In this paper, we derive some Fourier transforms of confluent hypergeometric functions. We give generalizations of several well-known results involving Fourier transforms of gamma functions. In particular, the generalizations include some Ramanujan's remarkable formulas.

Extended Conformable K-Hypergeometric Function and Its Application

The extended conformable k-hypergeometric function finds various applications in physics due to its ability to describe complex mathematical relationships arising in different physical scenarios. Here are a few instances of its uses in physics, including nuclear physics, fluid dynamics, quantum mechanics, and astronomy. The main objectives of this paper are to introduce the extended conformable k-hypergeometric and confluent hypergeometric functions by utilizing the new definition of the α,k-beta function and studying its important properties, like integral representation, summation formula, derivative formula, transform formula, and generating function. Also, introduce the extension of the Riemann–Liouville fractional derivative and establish some results related to the newly defined fractional operator, such as the Mellin transform and relations to extended α,k-hypergeometric functions.

Single-Shot Factorization Approach to Bound States in Quantum Mechanics

Single-Shot Factorization Approach to Bound States in Quantum Mechanics

Ground state of the Gross–Pitaevskii equation with a harmonic potential in the energy-critical case

Ground state of the energy-critical Gross–Pitaevskii equation with a harmonic potential can be constructed variationally. It exists in a finite interval of the eigenvalue parameter. The supremum norm of the ground state vanishes at one end of this interval and diverges to infinity at the other end. We explore the shooting method in the limit of large norm to prove that the ground state is pointwise close to the Aubin–Talenti solution of the energy-critical wave equation in near field and to the confluent hypergeometric function in far field. The shooting method gives the precise dependence of the eigenvalue parameter versus the supremum norm.

Generation of hyper-bunched light by single Gaussian and non-Gaussian scattering processes

We derive theoretically that hyper-bunched light with a central normalized second-order correlation coefficient of six can be realized by a single Gaussian scattering process of parametric down conversion (PDC) light with a central normalized second-order correlation coefficient of three. The Gaussian scattering process is realized by a rotating ground-glass diffuser. We show that the photon counting probability distribution in this case obeys a Tricomi confluent hypergeometric function U[1+n,3/2,1/⟨n⟩] dependence. Furthermore, we also study non-Gaussian light-scattering probabilities that together with the different impinging light statistics give rise to new photon statistics accompanied by a variety of new values of the second-order correlation coefficient of the scattered light. These theoretical calculations suggest experiments using twin photons from a PDC process and characterizing their photon statistics properties before and after the scattering at the rotating diffuser. These investigations contribute to a more comprehensive understanding of the scattering process, the generated light, and new applications.

Bi-Concave Functions Connected with the Combination of the Binomial Series and the Confluent Hypergeometric Function

In this article, we first define and then propose to systematically study some new subclasses of the class of analytic and bi-concave functions in the open unit disk. For this purpose, we make use of a combination of the binomial series and the confluent hypergeometric function. Among some other properties and results, we derive the estimates on the initial Taylor-Maclaurin coefficients |a2| and |a3| for functions in these analytic and bi-concave function classes, which are introduced in this paper. We also derive a number of corollaries and consequences of our main results in this paper.

Some Systems of PDE Associated with the Multiple Confluent Hypergeometric Functions and Their Applications

Some Systems of PDE Associated with the Multiple Confluent Hypergeometric Functions and Their Applications

The uniform asymptotic method “saddle point near an end point” revisited

We continue the program initiated in [López & all, 2009–2011] to simplify asymptotic methods for integrals: in this paper we revise the uniform method “saddle point near an end point”. We obtain a more systematic version of this uniform asymptotic method where the computation of the coefficients of the asymptotic expansion is remarkably simpler than in the classical method. On the other hand, as in the standard method, the asymptotic sequence is given in terms of parabolic cylinder functions. New asymptotic expansions of the confluent hypergeometric functions M(c,x/α+c+1,x) and U(c,αx+c+1,x) for large x, c fixed, uniformly valid for α∈(0,∞), are given as an illustration.

On some identities for confluent hypergeometric functions and Bessel functions

Mathematical functions, which often appear in mathematical analysis, are referred to as special functions and have been studied over hundreds of years. Many books and dictionaries are available that describe their properties and serve as a foundation of current science. In this paper, we find a new integral representation of the Whittaker function of the first kind and show a relevant summation formula for Kummer’s confluent hypergeometric functions. We also perform the specifications of our identities to link to known and new results.

IMPACT OF POROUS AND MAGNETIC DISSIPATION ON DISSIPATIVE FLUID FLOW AND HEAT TRANSFER IN THE PRESENCE OF DARCY-BRINKMAN POROUS MEDIUM

In this article, the boundary layer flow of an electrically-conducting fluid through a porous medium attached with a radiative permeable stretching sheet is analyzed. Following the Brinkman theory, an extended Darcy model (Darcy-Brinkman model) is utilized for the model momentum equation. Heat transfer analysis is also performed in the presence of viscous and Joule dissipation. Moreover, in the modeling of the energy equation, the effects of internal heating resulting from the mechanical effort required to squeeze out the fluid through the porous medium are also included in porous dissipation. Suitable dimensionless variables are introduced to convert the governing boundary layer equations into a dimensionless form, which are then converted into self-similar, nonlinear ordinary differential equations by utilizing similarity transformations. The exact solution of the nonlinear self-similar momentum equation is obtained in the form of the exponential function. In contrast, the solution of the energy equation is computed through the Laplace transform technique in the form of Kummer confluent hypergeometric functions. Effects of involved physical parameters on the momentum boundary layer (MBL), thermal boundary layer (TBL), wall shear stress, and local Nusselt number are explored through graphs and tables. Moreover, the slope linear regression (SLR) technique is used to calculate the rate of decrease/increase in shear stress and the rate of heat transfer at the boundary. The velocity and momentum boundary layer decreases for large values of porosity parameter and increases by increasing the viscosity ratio. The shear stress increases by increasing the porosity parameter, Hartman number, and suction parameter, while the opposite effect is examined with increasing values of viscosity ratio parameter. Heat transfer rate also enhances by increasing the Brinkman viscosity ratio parameter and wall suction velocity.

Asymptotic analysis for confluent hypergeometric function in two variables given by double integral

We study an integrable connection with irregular singularities along a normally crossing divisor. The connection is obtained from an integrable connection of regular singular type by a confluence, and has irregular singularities along \(x=\infty\) and \(y=\infty\). Solutions are expressed by a double integral of Euler type with resonances among the exponents in the integrand. We specify twisted cycles that give main asymptotic behaviors in sectorial domains around \((\infty,\infty)\). Then we obtain linear relations among the twisted cycles, and get an explicit expression of the Stokes multiplier. The methods to derive the asymptotic behaviors for double integrals and to get linear relations among twisted cycles in resonant case, which we developed in this paper, seem to be new.

ON SOME EXTENDED SPECIAL FUNCTIONS WITH THE BI-FOX-WRIGHT FUNCTION KERNEL

In the present article, our main goal is to study the new extended beta, Gauss and confluent hypergeometric functions defined by using the two Fox-Wright functions as their kernel. Some applications of the new extended beta function to distribution theory are also discussed.

Geometric Nature of Special Functions on Domain Enclosed by Nephroid and Leminscate Curve

In this work, the geometric nature of solutions to two second-order differential equations, zy′′(z)+a(z)y′(z)+b(z)y(z)=0 and z2y′′(z)+a(z)y′(z)+b(z)y(z)=d(z), is studied. Here, a(z), b(z), and d(z) are analytic functions defined on the unit disc. Using differential subordination, we established that the normalized solution F(z) (with F(0) = 1) of above differential equations maps the unit disc to the domain bounded by the leminscate curve 1+z. We construct several examples by the judicious choice of a(z), b(z), and d(z). The examples include Bessel functions, Struve functions, the Bessel–Sturve kernel, confluent hypergeometric functions, and many other special functions. We also established a connection with the nephroid domain. Directly using subordination, we construct functions that are subordinated by a nephroid function. Two open problems are also suggested in the conclusion.