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Two integral transforms involving Gaussian hypergeometric functions in the Kernel, which are generalization of classical Riemann-Liouville and Erdeyi-Kober fractional integral are considered and Applied on the Bessel functions of the first kind and thus two main results are derived, which are given in the form of two theorems. These two main results are obtained in terms of generalized wright functions, which are also expressed in terms of generalized hypergeometric functions, Corollaries & special cases are also considered. [ABSTRACT FROM AUTHOR]

The paper considers a function introduced by the authors, which depends on one complex variable, two real variables, and one more argument, which defines a trivial or proper subgroup of a three-dimensional proper Lorentz group, which, therefore, is a real number or a pair of real numbers. In this case, the first three arguments define representation spaces and basis functions in these spaces. It is shown that its particular values can be expressed via the Coulomb wave functions or Appell's hypergeometric function . The resulting formula for the transformation of the function is used to derive a continual addition theorem for this function and calculate the one-dimensional Fourier–Mellin-type integral transforms of the product of two Coulomb functions; its result is expressed via the function . [ABSTRACT FROM AUTHOR]

The paper considers a function introduced by the authors, which depends on one complex variable, two real variables, and one more argument, which defines a trivial or proper subgroup of a three-dimensional proper Lorentz group, which, therefore, is a real number or a pair of real numbers. In this case, the first three arguments define representation spaces and basis functions in these spaces. It is shown that its particular values can be expressed via the Coulomb wave functions or Appell's hypergeometric function . The resulting formula for the transformation of the function is used to derive a continual addition theorem for this function and calculate the one-dimensional Fourier–Mellin-type integral transforms of the product of two Coulomb functions; its result is expressed via the function . [ABSTRACT FROM AUTHOR]

We have examined the approximate lN-1-state solutions of the N-dimensional Schrödinger equation for a particle interacting with the Hellmann plus Kratzer potential. In hyperspherical coordinate system, we have constructed the bound state energy equation and the wavefunctions expressed by the hypergeometric function via the asymptotic iteration approach in detail. When considered the special cases of parameters in Hellmann plus Kratzer potential, this potential turns into several potential models. In this connection, the non-relativistic energy spectra for the modified Kratzer, Yukawa, Coulomb and Hellmann potentials in approximate analytic form have been obtained in hyperspherical coordinates. We have presented the numerical energy eigenvalues for the Hellmann, Yukawa and Coulomb potentials in N = 3 dimensions. Our present results provide an appropriate test of the accuracy of asymptotic iteration formalism. [ABSTRACT FROM AUTHOR]

Kimura and Takano showed that taking limits of regular elements of 𝔤 𝔩 (n) corresponds to the process of confluence of Aomoto–Gel'fand systems. We introduce a hypergeometric system associated with a principal nilpotent p -tuple, and, by using the principal nilpotent p -tuple, we directly deform a hypergeometric system of Gauss type into that of Airy type. Moreover, we explicitly describe the deformation. [ABSTRACT FROM AUTHOR]

Abstract (1) We consider a nonlinear generalization of the isotonic oscillator with an asymmetric potential. (2) Using a symmetrization principle we construct a symmetric potential. (3) The period function in this potential has the same value as in the original asymmetric potential. (4) It is amplitude dependent and expressible in terms of the hypergeometric function. Highlights • An exact analytical formula for the time period is calculated which reduces to that of the linear harmonic oscillator for the appropriate value of the relevant parameter. • A symmetrization argument is used to transform the potential such that the time period in the transformed symmetric potential matches that in the original asymmetric potential. [ABSTRACT FROM AUTHOR]

A new solution satisfying the Karmarkar condition is presented here. We were first to have discovered a hypergeometric function metric potential representing embedding class I spacetime. This new solution yields finite values of metric potentials, density, pressure, redshift, etc. and hence a non-singular solution. The solution is well behaved with respect to the parameter n = 12 to n = 24 corresponding to a stable configuration of mass 2.01M⊚ and radius 9.1 km. The internal properties of the solution are very different for n = 12 to n = 24; however, the total mass and radius is independent of the parameter n. The energy conditions are also holds good by the solution which thus can represent a physically viable matter distribution. The equilibrium condition and stability are also discussed through TOV-equation, cracking method and Γ > 4/3. The static stability criterion is also well satisfied and the turning point corresponds to 4.46M⊚ for a radius of 9.1 km. [ABSTRACT FROM AUTHOR]

Inclusion of coulomb interaction emerges with the complexity of either convergence of integrals or separation of variables of Schrödinger equations. For an N -electron system, interaction terms grow by N ( N -1)/2 factors. Therefore, 2- e system stands as fundamental basic unit for generalized N - e systems. For the first time, we have evaluated e – e correlations in very simple and absolutely terminating finite summed hypergeometric series for 2-D double carrier parabolic quantum dot in both zero and arbitrary non-zero magnetic field (symmetric gauge) and have appraised these integrals in variational methods. The competitive role among confinement strength, magnetic field, mass of the carrier and dielectric constant of the medium on energy level diagram, level-spacing statistics, heat capacities ( C v at 1 K) and magnetization ( T ∼ (0–1)K) is studied on systems spanning over wide range of materials (GaAs,Ge,CdS,SiO 2 and He, etc). We have also constructed an exact theory for generalized correlated N - e 2-D quantum dots via multi-pole expansion but for the sake of compactness of the article we refrain from data. [ABSTRACT FROM AUTHOR]

In 1946, Magnus presented an addition theorem for the confluent hypergeometric function of the second kindUwith argumentx+yexpressed as an integral of a product of twoU's, one with argumentxand another with argumenty. We take advantage of recently obtained asymptotics forUwith large complex first parameter to determine a domain of convergence for Magnus' result. Using well-known specializations ofU, we obtain corresponding integral addition theorems with precise domains of convergence for modified parabolic cylinder functions, and Hankel, Macdonald, and Bessel functions of the first and second kind with order zero and one. [ABSTRACT FROM PUBLISHER]

In 2013, Lemke Oliver classified all eta-quotients which are theta functions. In this paper, we unify the eta-theta functions by constructing mock modular forms from the eta-theta functions with even characters, such that the shadows of these mock modular forms are given by the eta-theta functions with odd characters. In addition, we prove that our mock modular forms are quantum modular forms. As corollaries, we establish simple finite hypergeometric expressions which may be used to evaluate Eichler integrals of the odd eta-theta functions, as well as some curious algebraic identities. [ABSTRACT FROM AUTHOR]

Kuznetsov and co-authors in 2011‒14 introduced the family of hypergeometric Lévy processes. They appear naturally in the study of fluctuations of stable processes when one analyses stable processes through the theory of positive self-similar Markov processes. Hypergeometric Lévy processes are defined through their characteristic exponent, which, as a complex-valued function, has four independent parameters. In 2014 it was shown that the definition of a hypergeometric Lévy process could be taken to include a greater range of the aforesaid parameters than originally specified. In this short article, we push the parameter range even further. [ABSTRACT FROM AUTHOR]

Hypergeometric functions of matrix arguments occur frequently in multivariate statistical analysis. In this paper, we define and study extended forms of Gauss and confluent hypergeometric functions of matrix arguments and show that they occur naturally in statistical distribution theory. [ABSTRACT FROM AUTHOR]

The quantum mechanical problem for a spin ½ particle in external Coulomb potential, reduced to a system of two first-order differential equations, is reconsidered from the point of view of solving this system by using the Heun function theory. It is shown that, besides the standard approach of solving the problem in terms of confluent hypergeometric functions, there are several other possibilities, which rely on using the confluent Heun functions. We consider two new methods to construct the solutions of the problem: the first implies that only one component of the pair of relevant functions is expressed in terms of the Heun functions, and in the second approach both functions of the system are expressed in terms of the Heun functions. In this context, certain relations between the two classes of involved functions are established. It is shown that all the considered cases lead to the same energy spectrum, which validates the correctness of the approaches. [ABSTRACT FROM AUTHOR]

We propose a mechanism of ultrahigh-Q metallic nanocavity resonances that involves an efficient loss-compensation scheme favorable for room-temperature operation. We theoretically show that surface plasmon-polaritons excited on the entrance and exit interfaces of a metallic nanocavity array efficiently transfer external optical gain to the cavity modes by inducing resonantly-amplified intracavity feedback. Surprisingly, the modal gain in the nanocavity with the externally amplified feedback is inversely proportional to the cavity length as opposed to conventional optical cavity amplifiers requiring longer cavities for higher optical gain. Utilizing this effect, we numerically demonstrate room-temperature nanocavity resonance Q-factor exceeding 104 in a 25-nm-wide silver nanoslit array. The proposed mechanism provides a highly efficient plasmonic amplification process particularly for subwavelength plasmonic cavities which are essential components in active nanoplasmonic devices. [ABSTRACT FROM AUTHOR]

The article presents as study about trajectories of principal stresses in the plane-stress state of material obeying the Tresca and Coulomb-Mohr yield conditions. The study considered the plane stress state of an ideally plastic material obeying the Tresca yield condition. It also introduced the curvilinear system of coordinates (q1, q2), the coordinate lines of which are the trajectories of the principal stresses o1 and o2 , respectively.

Owing to a strong Coulomb repulsion, heavy electron superconductors mostly have anisotropic gap functions which have nodes for certain directions in the momentum space. Since the nodal structure is closely related to the pairing mechanism, its experimental determination is of primary importance. This article discusses the experimental methods of the gap determination by bulk heat capacity measurements in a rotating magnetic field. The basic idea is based on the fact that the quasiparticle density of states in the vortex state of nodal superconductors is field and direction dependent. We present our recent experimental results of the field-orientation dependence of the heat capacity in heavy fermion superconductors CeTIn5 (T  =  Co, Ir), UPt3, CeCu2Si2, and UBe13 and discuss their gap structures. [ABSTRACT FROM AUTHOR]

A derivation of the ionization rate for a hydrogen atom in its ground state (or a hydrogen-like positive ion) in a strong linearly polarized laser field is presented. The derivation utilizes the famous Keldysh probability amplitude in the length gauge (in the dipole approximation) and without Coulomb effects in the final state of the ionized electron. No further approximations are made, because the amplitude has been expanded in the double Fourier series in a time domain (with the help of the generalized Bessel functions). Thus, our theory has no other limitations that are characteristic of the original Keldysh theory. We compare our ‘exact’ theory with the original Keldysh one by studying photoionization energy spectra and total ionization rates. We show a breakdown of the original Keldysh theory for higher frequencies (when the photon energy approaches the binding energy). We also compare our theory with the analogous result in the velocity gauge. In the barrier-suppression regime, the ‘exact’ Keldysh theory gives results which are close to the well-known empirical formula and close to some other numerical or theoretical results. Numerous comparisons of total ionization rates are limited to photons of energies lower or much lower than the binding energy of the atom. [ABSTRACT FROM AUTHOR]

A dynamically screened three-Coulomb-wave (DS3C) method is applied to study the single ionization of magnesium by electron impact. Triple differential cross sections (TDCS) are calculated in doubly symmetric geometry at incident energies of 13.65, 17.65, 22.65, 27.65, 37.65, 47.65, 57.65, and 67.65 eV. Comparisons are made with experimental data and theoretical predictions from a three-Coulomb-wave function (3C) approach and distorted-wave Born approximation (DWBA). The overall agreement between the predictions of the DS3C model and the DWBA approach with the experimental data is satisfactory. [ABSTRACT FROM AUTHOR]