Terms Of Hypergeometric Functions Research Articles

Quasinormal modes of charged fermions in linear dilaton black hole spacetime: exact frequencies

We study charged massless fermionic perturbations in the background of $4$-dimensional linear dilaton black holes in Einstein-Maxwell-dilaton theory with double Liouville-type potentials. We present the analytical fermionic quasinormal modes, whose Dirac equations are solved in terms of hypergeometric functions. We also discuss the stability of these black holes under the charged fermionic perturbations.

Аналитическое решение уравнения Даффина - Кеммера - Петьо для суммы потенциала Маннинга - Розена и класс Юкавы

This paper presents an analytical bound-state solution to the Duffin - Kemmer - Petiau equation for the new putative combined Manning - Rosen and Yukawa class potentials. Using the developed scheme to approximate and overcome the difficulties arising in the centrifugal part of the potential, the bound-state solution of the modified Duffin - Kemmer - Petiau equation is found. Analytical expressions of energy eigenvalue and the corresponding radial wave functions are obtained for an arbitrary value of the orbital quantum number l . Also, eigenfunctions are expressed in terms of hypergeometric functions. It is shown that energy levels and eigenfunctions are quite sensitive to the choice of radial and orbital quantum numbers.

Integrable Abel equation and asymptotics of symmetry solutions of Korteweg-de Vries equation

We provide a general solution for a first order ordinary differential equation with a rational right-hand side, which arises in constructing asymptotics for large time of simultaneous solutions of the Korteweg-de Vries equation and the stationary part of its higher non-autonomous symmetry. This symmetry is determined by a linear combination of the first higher autonomous symmetry of the Korteweg-de Vries equation and of its classical Galileo symmetry. This general solution depends on an arbitrary parameter. By the implicit function theorem, locally it is determined by the first integral explicitly written in terms of hypergeometric functions. A particular case of the general solution defines self-similar solutions of the Whitham equations, found earlier by G.V. Potemin in 1988. In the well-known works by A.V. Gurevich and L.P. Pitaevsky in early 1970s, it was established that these solutions of the Whitham equations describe the origination in the leading term of non-damping oscillating waves in a wide range of problems with a small dispersion. The result of this article supports once again an empirical rule saying that under various passages to the limits, integrable equations can produce only integrable, in certain sense, equations. We propose a general conjecture: integrable ordinary differential equations similar to that considered in the present paper should also arise in describing the asymptotics at large times for other symmetry solutions to evolution equations admitting the application of the method of inverse scattering problem.

О построении функции Римана-Адамара для трехмерного уравнения Бианки

In this paper, we state the Darboux problem and give the definition of the Riemann–Hadamard function for a third-order equation with dominant partial derivative (the Bianchi equation). Basing on the possibility of explicitly representing the Riemann function for a certain class of Bianchi equations of the third order, which are equivalent with respect to function, we establish sufficient conditions for coefficients of the Bianchi equation that provide the construction of the Riemann–Hadamard function in terms of hypergeometric functions.

Pseudospin and spin symmetry in the relativistic generalized Woods-Saxon potential including Coulomb-like tensor potential

The Dirac Equation is solved approximately for relativistic generalized Woods-Saxon potential including Coulomb-like tensor potential in exact pseudospin and spin symmetry limits. The bound states energy eigenvalues are found by using wavefunction boundary conditions, and corresponding radial wavefunctions are obtained in terms of hypergeometric function. Some numerical examples are given for the dependence of bound states energy eigenvalues on quantum numbers and potential parameters.

Solution of the Ultra Generalized Exponential–Hyperbolic Potential in Multi-dimensional Space

In this work, we proposed ultra generalized exponential–hyperbolic potential (UGEHP) and derived various well known exponential–hyperbolic type potentials by setting parameters in UGEHP and using approximation suggested by Greene–Aldrich. The bound state solutions of the multi (D)-dimensional Schrodinger equation for UGEHP have been presented using the parametric Nikiforov–Uvarov method. The approximate analytical bound state energy eigenvalues and the corresponding un-normalized eigenfunctions expressed in terms of hypergeometric functions were obtained. We also investigated the rotational vibrational (RV) partition function from the eigenvalue of UGEHP. By the setting parameters, we obtained eigenvalue spectrum and RV partition function for the screened cosine Kratzer potential, screened Kratzer potential, attractive radial potential, quadratic exponential-type potential, Manning Rosen with class of Yukawa potential and Yukawa potential, class of Yukawa potential, mixed class of Yukawa potential, quantum interaction potential or Hulthen–Yukawa inversely quadratic potential, Hulthen plus inversely quadratic exponential Mie-type potential and Hulthen plus exponential Coulombic potential with centrifugal potential barrier. We studied the behavior of energy eigenvalues and RV partition function for the UGEHP. We computed and tabulated numerical results for $$CO, NO, I_2$$ , HCl and LiH diatomic molecules and compared with numerical results available in literature for same diatomic molecules.

Schrödinger’s ants: a continuous description of Kirman’s recruitment model

We show how the approach to equilibrium in Kirman’s ants model can be fully characterized in terms of the spectrum of a Schrödinger equation with a Pöschl–Teller (tan2) potential. Among other interesting properties, we have found that in the bimodal phase where ants visit mostly one food site at a time, the switch time between the two sources only depends on the ‘spontaneous conversion’ rate and not on the recruitment rate. More complicated correlation functions can be computed exactly, and involve higher and higher eigenvalues and eigenfunctions of the Schrödinger operator, which can be expressed in terms of hypergeometric functions.

THE BIVARIATE EXTENSION OF AMOROSO DISTRIBUTION

This paper introduces the bivariate extension of the amoroso distribution and its density function is expressed in terms of hyper-geometric function. The standard amoroso distribution, cumulative distribution functions, conditional distributions, and its moments are also derived. The Product moments, Co-variance, correlations, and Shannon’s differential entropy are also shown. Moreover, the generating functions such as moment, Cumulant, Characteristic functions are expressed in Fox-wright function, and the Survival, hazard, and Cumulative hazard functions are also computed. The special cases of the bivariate amoroso distribution are also discussed and nearly 780 bivariate mixtures of distributions can be derived. Finally, the two-dimensional probability surfaces are visualized for the selected special cases and we also showed the estimation of parameters by the method of maximum likelihood approach, and the constrained maximum likelihood approach is also computed by using Non-linear Programming with a numerical application

On the Volterra-Type Fractional Integro-Differential Equations Pertaining to Special Functions

In this article, we apply an integral transform-based technique to solve the fractional order Volterra-type integro-differential equation (FVIDE) involving the generalized Lorenzo-Hartely function and generalized Lauricella confluent hypergeometric function in terms of several complex variables in the kernel. We also investigate and introduce the Elazki transform of Hilfer-derivative, generalized Lorenzo-Hartely function and generalized Lauricella confluent hypergeometric function. In this article, we have established three results that are present in the form of lemmas, which give us new results on the above mentioned three functions, and by using these results we have derived our main results that are given in the form of theorems. Our main results are very general in nature, which gives us some new and known results as a particular case of results established here.

Self-Similar Solutions of Some Model Degenerate Partial Differential Equations of the Second, Third, and Fourth Order

When studying boundary value problems for some partial differential equations arising in applied mathematics, we often have to study the solution of a system of partial differential equations satisfied by hypergeometric functions and find explicit linearly independent solutions for the system. In this study, we construct self-similar solutions of some model degenerate partial differential equations of the second, third, and fourth order. These self-similar solutions are expressed in terms of hypergeometric functions.

Degenerate Analogues of Euler Zeta, Digamma, and Polygamma Functions

In recent years, much attention has been paid to the role of degenerate versions of special functions and polynomials in mathematical physics and engineering. In the present paper, we introduce a degenerate Euler zeta function, a degenerate digamma function, and a degenerate polygamma function. We present several properties, recurrence relations, infinite series, and integral representations for these functions. Furthermore, we establish identities involving hypergeometric functions in terms of degenerate digamma function.

Energies Spectra and Thermodynamic Properties of Hyperbolic Pöschl–Teller Potential (HPTP) Model

We solved the Dirac equation using modified factorization method with hyperbolic Poschl–Teller potential model. Relativistic and non-relativistic ro-vibrational energy spectra were obtained and numerical solutions for four diatomic molecules $$\left( {{\text{HCl}},\;{\text{H}}_{2} ,\;{\text{CO}}\;{\text{and}}\;{\text{LiH}}} \right)$$ obtained also. The eigenfunction for this potential has been obtained in terms of hypergeometric function. The energy variations with different potential parameters and quantum numbers of the hyperbolic Poschl–Teller potential are also plotted. In addition, we evaluate the vibrational partition function and other thermodynamic properties such as vibrational internal energy, vibrational free energy, vibrational entropy and vibrational specific heat capacity, in terms of temperature and upper bound vibration quantum number. Our results are consistent with those found in the available literature.

Hypergeometric presentation for one-loop contributing to H → Z γ

Abstract In this paper, new analytic formulas for one-loop contributing to Higgs decay channel $H \rightarrow Z\gamma$ are presented in terms of hypergeometric functions. The calculations are performed by following the technique for tensor one-loop reduction developed in [A. I. Davydychev, Phys. Lett. B 263 (1991) 107]. For the first time, one-loop form factors for the decay process are shown which are valid at arbitrary space–time dimension $d$.

Binomial polynomials mimicking Riemann's zeta function

ABSTRACT The (generalized) Mellin transforms of Gegenbauer polynomials have polynomial factors , whose zeros all lie on the ‘critical line’ (called critical polynomials). The transforms are identified in terms of combinatorial sums related to H. W. Gould's S:4/3, S:4/2, and S:3/1 binomial coefficient forms. Their ‘critical polynomial’ factors are then identified in terms of hypergeometric functions. Furthermore, we extend these results to a one-parameter family of critical polynomials that possess the functional equation . Normalization yields the rational function whose denominator has singularities on the negative real axis. Moreover, as along the positive real axis, from below. For the Chebyshev polynomials, we obtain the simpler S:2/1 binomial form, and with the nth Catalan number, we deduce that and yield odd integers. The results touch on analytic number theory, special function theory, and combinatorics.

Uniformly convergent expansions for the generalized hypergeometric functions p –1 Fp and pFp

ABSTRACT We derive a convergent expansion of the generalized hypergeometric function in terms of the Bessel functions 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 in terms of the confluent hypergeometric functions that holds uniformly in any right half-plane. For both functions, we make a further step forward and give convergent expansions in terms of trigonometric, exponential and rational functions that hold uniformly in the same domains. For all four expansions we present explicit error bounds. The accuracy of the approximations is illustrated by some numerical experiments.

Eigensolution and various properties of the screened cosine Kratzer potential in D dimensions via relativistic and non-relativistic treatment

In this article, we proposed screened cosine Kratzer potential (SCKP). Using approximation suggested by Greene–Aldrich, the approximate bound-state solutions of the D-dimensional Klein–Gordon equation for SCKP have been obtained via the generalized Nikiforov–Uvarov method. The bound-state energy eigenvalues and the corresponding normalized wave functions expressed in terms of hypergeometric functions were obtained. From the proposed SCKP model, we recovered different potentials such as screened Kratzer potential, standard Kratzer potential, generalized cosine Yukawa potential, screened Coulomb potential, Coulomb potential, inversely quadratic Yukawa potential and its corresponding energy eigenvalues from obtained energy eigenvalues of the SCKP in both the relativistic and non-relativistic regime. We obtained rotational–vibrational energy for few heterogeneous (LiH, HCl, NO) and homogeneous (H $$_2$$ , I $$_2$$ , O $$_2$$ ) diatomic molecules in three dimensions. The numerical results obtained for LiH, HCl, NO and I $$_2$$ , O $$_2$$ diatomic molecules are in very good agreement with the results previously obtained by others. We obtained various properties of the SCKP such as thermodynamical properties (partition function, vibrational mean energy, vibrational mean free energy, vibrational specific heat capacity and vibrational entropy, information energy $$E(\gamma )$$ , Tsallis entropy T(q), Renyi entropy R(q), Shannon entropy $$ S(\gamma )$$ and Fisher information entropy $$I(\gamma )$$ ) using energy eigenvalues and eigenfunctions, the expressions for the expectation values of the square of inverse of position $$1/r^2$$ , inverse of position 1/r, kinetic energy T, and square of momentum $$p^2$$ via Hellmann–Feynman theorem. We also presented quarkonium mass spectroscopy using energy eigenvalues of the SCKP.

Pair creation in curved Snyder space

We study the problem of pair creation of scalar particles by an electric field in curved Snyder space. We find exact solutions for the Klein–Gordon equation with a constant electric field in terms of hypergeometric functions. Then we calculate the pair creation probability and the number of created pairs of particles through Bogoliubov transformation technique.

Expression of the Holtsmark function in terms of hypergeometric $$_{2}F_2$$ and Airy Bi functions

The Holtsmark distribution has applications in plasma physics, for the electric-microfield distribution involved in spectral line shapes for instance, as well as in astrophysics for the distribution of gravitating bodies. It is one of the few examples of a stable distribution for which a closed-form expression of the probability density function is known. However, the latter is not expressible in terms of elementary functions. In the present work, we mention that the Holtsmark probability density function can be expressed in terms of hypergeometric function $$_{2}F_2$$ and of Airy function of the second kind Bi and its derivative. The new formula is simpler than the one proposed by Lee involving $$_{2}F_3$$ and $$_{3}F_4$$ hypergeometric functions.

Fundamental Solutions for a Class of Multidimensional Elliptic Equations with Several Singular Coefficients

The main result of the present paper is the construction of fundamental solutions for a class of multidimensional elliptic equations with several singular coefficients. These fundamental solutions are directly connected with multiple hypergeometric functions and the decomposition formula is required for their investigation which would express the multivariable hypergeometric function in terms of products of several simpler hypergeometric functions involving fewer variables. In this paper, such a formula is proved instead of a previously existing recurrence formula.The order of singularity and other properties of the fundamental solutions that are necessary for solving boundary value problems for degenerate second-order elliptic equations are determined

Schrödinger’s Ants: A Continuous Description of Kirman’s Recruitment Model

We show how the approach to equilibrium in Kirman’s ants model can be fully characterized in terms of the spectrum of a Schrodinger equation with a Poschl-Teller (tan2) potential. Among other interesting properties, we have found that in the bimodal phase where ants visit mostly one food site at a time, the switch time between the two sources only depends on the “spontaneous conversion” rate and not on the recruitment rate. More complicated correlation functions can be computed exactly, and involve higher and higher eigenvalues and eigenfunctions of the Schrodinger operator, which can be expressed in terms of hypergeometric functions.