Hypergeometric Form Research Articles

New results of unified Chebyshev polynomials

<abstract><p>This paper presents a new approach for the unified Chebyshev polynomials (UCPs). It is first necessary to introduce the three basic formulas of these polynomials, namely analytic form, moments, and inversion formulas, which will later be utilized to derive further formulas of the UCPs. We will prove the basic formula that shows that these polynomials can be expressed as a combination of three consecutive terms of Chebyshev polynomials (CPs) of the second kind. New derivatives and connection formulas between two different classes of the UCPs are established. Some other expressions of the derivatives of UCPs are given in terms of other orthogonal and non-orthogonal polynomials. The UCPs are also the basis for additional derivative expressions of well-known polynomials. A new linearization formula (LF) of the UCPs that generalizes some well-known formulas is given in a simplified form where no hypergeometric forms are present. Other product formulas of the UCPs with various polynomials are also given. As an application to some of the derived formulas, some definite and weighted definite integrals are computed in closed forms.</p></abstract>

Hiemenz stagnation point flow with computational modelling of variety of boundary conditions

This work explains the flow of a G-H2O nanofluid under the special case of MHD stagnation point flow, and the detailed investigation of the Navier’s stokes equations extracted analytically. The main methodology is given work of PDEs is converted into ODEs using the appropriate similarity transformations. The momentum equations solved analytically to derive the solution domain, the impact of thermal radiation is seen in energy equation, four different scenarios are used to solve the energy equation. Aim of the present work is to study the theoretical analysis and it can be discussed for dual nature behavior by providing different physical parameters, these parameters control the domain, momentum and heat transpiration. In the heat transfer analysis solutions are derived in terms of incomplete gamma function and confluent hypergeometric form. The current work is examined using graphene nanoparticles, and the value of Pr is fixed at 6.2. The present problem is the benchmark solution for the results and it is significance in industrial and technological applications in fluid-based systems involving shrinkable/stretchable materials. At the end we get Velocity decreases with increases of VC for upper branch of solution and increases with increases of VC for lower branch of solution in the case shrinking sheet.

An MHD Casson fluid flow past a porous stretching sheet with threshold Non-Fourier heat flux model

The present paper investigates the magnetohydrodynamics Casson fluid flow with porous medium. The Cu-water nanofluid is used to investigate further analytical results. Thermal radiation is considered along with the Cattaneo-Christov heat flux model and then solved analytically, then expresses the closed forms analytical solution in terms of two variables of hypergeometric function. The special form of Appell hypergeometric solutions are closed form of analytical solutions. The results of normalized shear stress at the wall and temperature profile, and rate of heat flux can be analyzed with suitable parameters, Viz., relaxation time parameter, Prandtl number, Radiation parameter, Magnetic parameter, and so on. The newness of the current work's novelty is to explain a solution to issues with stretching sheets on the basis of Appell hypergeometric form and also investigation takes place on a time relaxation parameter. The shear stress fηη0 is also calculated analytically. This problem has many industrial applications and engineering processes, namely glass-fiber production, extrusion of rubber sheets and so on. Finding that an increase in the thermal relaxation parameter and Prandtl number keeps the fluid temperature constant is the primary physical application.

HYPERGEOMETRIC FORM OF (1 + x2)ib/2 exp(b tan-1 x) AND ITS APPLICATIONS

HYPERGEOMETRIC FORM OF (1 + x2)ib/2 exp(b tan-1 x) AND ITS APPLICATIONS

Hollow vortices of neutrino from Kerr black hole

In the present paper, a novel family of neutrino fields with a hollow distribution outside a rotating black hole is developed. To analyze the hollow neutrino field which excited from the Kerr black hole a theoretical method is proposed. The neutrinos wave function from the Kerr black hole is reported as hypergeometric form. Similar to the hollow in the core of the rotating black hole, the emitted neutrinos with a dark core are reported. The emitted neutrino beams are characterized by a tube of energy and angular momentum with a singularity at the center. The tubes have a symmetric axis that coincided with the symmetric axis of the rotating black hole. The number rate, emitted power, and angular momentum of the neutrinos near the rotation axis of the black hole vanish. Furthermore, the maximum rate is taken place near the equator plane of the black hole. As the black hole is rotating, its angular momentum is transferred to the emitted neutrinos, hence the helicity of the neutrinos wave function is excited.

An MHD Flow of Non-Newtonian Fluid Due to a Porous Stretching/Shrinking Sheet with Mass Transfer

An examination is carried out for three-dimensional incompressible viscoelastic fluid flow over a porous stretching/shrinking sheet with hybrid nanoparticles copper-alumina (Cu−Al2O3) in base fluid water (H2O). The uniform magnetic field of strength B0 is applied perpendicular to the fluid flow and considered the Navier slip. The mass transfer is considered with the chemical reaction rate. The governing equation for the defined flow forms the system of partial differential equations, which are then transformed into a system of ordinary differential equations via similarity transformations. The goal is to find the exact analytical solution, and the unique solution is determined by considering the boundary layer theory. Furthermore, the obtained system is solved to get the exact analytical solution for velocity and concentration fields in exponential form and in hypergeometric form, respectively. The exact solutions are obtained for velocity and temperature profiles, Skin friction, and Nusselt number. These findings are beneficial for future research in the present area. The parameters magnetic field, Inverse Darcy number, slip parameter, chemical reaction parameter, stretching/shrinking parameter, and viscoelastic parameter, influence the flow. The effect of these parameters on fluid velocity and concentration field will be analyzed through graphs. Skin friction and Nusselt number are also analyzed. This work found many applications in machining and manufacturing, solar energy, MHD flow meters and pumps, power generators, geothermal recovery, flow via filtering devices, chemical catalytic reactors, etc.

Analysis of a Stagnation Point Flow With Hybrid Nanoparticles over a Porous Medium

The unsteady stagnation-point flow of a hybrid nanofluid over a stretching/shrinking sheet embedded in a porous medium with mass transpiration and chemical reactions is considered. The momentum and mass transfer problems are combined to form a system of partial differential equations, which is converted into a set of ordinary differential equations via similarity transformation. These ordinary differential equations are solved analytically to obtain the solution for velocity and concentration profiles in exponential and hypergeometric forms, respectively. The concentration profile is obtained for four different cases namely constant wall concentration, uniform mass flux, general power law wall con-centration and general power law mass flux. The effect of different physical parameters such as Darcy number (Da-1), mass transpiration parameter (VC), stretching/shrinking parameter (d), chemical reaction parameter (β) and Schmidt number (SC) velocity and concentration profile is examined. Results show that, the axial velocity will decreases as the shrinking sheet parameter increases, regardless of whether the suction or injection case is examined. The concentration decreases with an increase in the shrinking sheet parameter and the chemical reaction rate parameter.

HYPERGEOMETRIC REPRESENTATIONS OF SOME MATHEMATICAL FUNCTIONS VIA MACLAURIN SERIES

In this paper, by using Maclaurin series of given mathematical function and expressing the coefficient of the general term of corresponding Maclaurin series in the form of Pochhammer symbols, we obtain the hypergeometric forms of following functions:

HYPERGEOMETRIC FORMS OF SOME MATHEMATICAL FUNCTIONS VIA DIFFERENTIAL EQUATION APPROACH

In this paper, by changing the independent and dependent variables in the suitable ordinary differential equations of first and second order, and comparing the resulting ordinary differential equations with standard ordinary differential equations of Leibnitz and Gauss, we obtain the hypergeometric forms of following functions:

On some hypergeometric solutions of the conformal Ward identities of scalar 4-point functions in momentum space

We discuss specific hypergeometric solutions of the conformal Ward identities (CWI’s) of scalar 4-point functions of primary fields in momentum space, in d spacetime dimensions. We determine such solutions using various dual conformal ansätze (DCA’s). We start from a generic dual conformal correlator, and require it to be conformally covariant in coordinate space. The two requirements constrain such solutions to take a unique hypergeometric form. They describe correlators which are at the same time conformal and dual conformal in any dimension. These specific ansätze also show the existence of a link between 3- and 4-point functions of a CFT for such class of exact solutions, similarly to what found for planar ladder diagrams. We show that in d = 4 only the box diagram and its melonic variants, in free field theory, satisfies such conditions, the remaining solutions being nonperturbative. We then turn to the analysis of some approximate high energy fixed angle solutions of the CWI’s which also in this case take the form of generalized hypergeometric functions. We show that they describe the behaviour of the 4-point functions at large energy and momentum transfers, with a fixed −t/s. The equations, in this case, are solved by linear combinations of Lauricella functions of 3 variables and can be rewritten as generalized 4K integrals. In both cases the CWI’s alone are sufficient to identify such solutions and their special connection with generalized hypergeometric systems of equations.

A study of thermodynamic properties of quadratic exponential-type potential in D-dimensions

We solved the Schrodinger equation with Quadratic Exponential-Type Potential (QEP) model in D-dimensions using the Modified factorization method. The energy eigenvalues and total wavefunctions were obtained in a Gauss hypergeometric form. The thermodynamic properties including vibrational partition function, vibrational mean energy, vibrational mean free energy and vibrational entropy have been calculated for the electronic state of (X1 Σ +g) Rubidium (Rb2) dimer. The QEP discussed can be applied extensively in Physics and Chemistry, especially in molecular dynamics.

Solutions of the radial Schrödinger equation in hypergeometric and discrete fractional forms

The purpose of this present paper is to obtain the hypergeometric and discrete fractional solutions of the radial Schrodinger equation by using the nabla discrete fractional calculus operator.

Formulas for Generalized Two-Qubit Separability Probabilities

To begin, we find certain formulas Q(k,α)=G1k(α)G2k(α), for k=-1,0,1,…,9. These yield that part of the total separability probability, P(k,α), for generalized (real, complex, quaternionic, etc.) two-qubit states endowed with random induced measure, for which the determinantal inequality ρPT>ρ holds. Here ρ denotes a 4×4 density matrix, obtained by tracing over the pure states in 4×(4+k)-dimensions, and ρPT denotes its partial transpose. Further, α is a Dyson-index-like parameter with α=1 for the standard (15-dimensional) convex set of (complex) two-qubit states. For k=0, we obtain the previously reported Hilbert-Schmidt formulas, with Q(0,1/2)=29/128 (the real case), Q(0,1)=4/33 (the standard complex case), and Q(0,2)=13/323 (the quaternionic case), the three simply equalling P(0,α)/2. The factors G2k(α) are sums of polynomial-weighted generalized hypergeometric functions pFp-1, p≥7, all with argument z=27/64=(3/4)3. We find number-theoretic-based formulas for the upper (uik) and lower (bik) parameter sets of these functions and, then, equivalently express G2k(α) in terms of first-order difference equations. Applications of Zeilberger’s algorithm yield “concise” forms of Q(-1,α), Q(1,α), and Q(3,α), parallel to the one obtained previously (Slater 2013) for P(0,α)=2Q(0,α). For nonnegative half-integer and integer values of α, Q(k,α) (as well as P(k,α)) has descending roots starting at k=-α-1. Then, we (Dunkl and I) construct a remarkably compact (hypergeometric) form for Q(k,α) itself. The possibility of an analogous “master” formula for P(k,α) is, then, investigated, and a number of interesting results are found.

Discrete fractional solutions of radial Schrödinger equation for Makarov potential

The fractional calculus that is a theory of integral and derivative with arbitrary order is an important subject of applied mathematics. This theory has extensive usage fields in science and engineering. Discrete mathematics, the study of finite structures, is one of the fastest growing areas in mathematics and optimization. Recently, many considerable scientific works were published on fractional calculus and discrete fractional calculus (DFC). The purpose of this paper is to obtain discrete fractional solutions of radial Schrödinger equation for Makarov potential by means of nabla DFC operator. Moreover, we introduce hypergeometric forms of these solutions.

New discrete orthogonal moments for signal analysis

The paper addresses some computational aspects and applications of the polynomial unbiased finite impulse response functions originally derived by Shmaliy as a new class of a one-parameter family of discrete orthogonal polynomials. We present two new explicit formulas to compute these polynomials directly. They are based on the discrete Rodrigues’ representation and hypergeometric form, respectively. These straightforward calculations lead us to propose another discrete signal representation in moment domain. Experimental results show a comparison between the proposed moments and discrete Chebyshev moments in terms of noise-free/noisy signal feature extraction and reconstruction error.

An elementary solution of Gessel's walks in the quadrant

Around 2000, Ira Gessel conjectured that the number of lattice walks in the quadrant N2, starting and ending at the origin (0,0) and taking their steps in {→,↗,←,↙} had a simple hypergeometric form. In the following decade, this problem was recast in the systematic study of walks with small steps (that is, steps in {−1,0,1}2) confined to the quadrant. The generating functions of such walks are archetypal solutions of partial discrete differential equations.A complete classification of quadrant walks according to the nature of their generating function (algebraic, D-finite or not) is now available, but Gessel's walks remained mysterious because they were the only model among the 23 D-finite ones that had not been given an elementary solution. Instead, Gessel's conjecture was first proved using an inventive computer algebra approach in 2008. A year later, the associated three-variate generating function was proved to be algebraic by a computer algebra tour de force. This was re-proved recently using elaborate complex analysis machinery. We give here an elementary and constructive proof. Our approach also solves other quadrant models (with multiple steps) recently proved to be algebraic via computer algebra.

Solutions of the Radial Component of the Fractional Schrödinger Equation Using N-Fractional Calculus Operator

Fractional calculus tecniques are used for the solutions of some classes of differential equations and fractional differential equations. One of the these tecniques is N-fractional calculus operator \(N^{\eta }\) method. We can obtain the fractional solutions differently from classical solutions by means of \(N^{\eta }\) method. In this study, we applied the \(N^{\eta }\) method to the radial component of the fractional Schrodinger equation. After, we obtained hypergeometric forms of the solutions.

Commutative vs. Noncommutative Space Statistical Properties of Two-Dimensional Harmonic Oscillator in Magnetic Field

The Schrodinger equation and its solutions are still a challenging subject in physics. Just as many other areas of science, where we have to solve a differential equation to obtain the required information, we have to first solve the building block of nonrelativistic quantum mechanics. In doing so, we have to use numerical and analytical techniques depending on the structure of the equation. In particular, the analytical approaches are attractive as they provide a deeper and more touchable insight into the physics of the problem [1–6]. Cooper et al. reviewed the theoretical formulation of supersymmetry quantum mechanics and discussed its applications in dealing with both relativistic and nonrelativistic equations of quantum mechanics [7]. Ciftci et al. used the asymptotic iteration method for finding solutions of the Schrodinger equation [8]. Slater considered a simplification of the Hartree–Fock method to analyze the related problems [9]. Stevenson applied the optimized perturbation theory to the field [10]. Dong et al. proposed the quasi-exact solutions of the Schrodinger equation via the ansatz technique which is a quasi-exact approach [11]. Another frequently used tool is the Nikiforov–Uvarov (NU) technique which transforms classes of equations of mathematical physics into hypergeometric form. A reason of recent renewed interests in the wave equations of quantum mechanics is the implications of fundamental theories such as string theory. To be more precise, the noncommutative (NC) formulations of quantum mechanics and the proposition of the so-called minimal length, has motivated many recent studies on the quantum equations. The NC formulation, which is the focus of the present work, originates from fundamental theories

Some multiple series identities and their hypergeometric forms

In this paper, we obtain solutions of some multiple series identities involving bounded multiple sequences. We also derive hypergeometric forms of these identities involving Kampé de Fériet double hypergeometric function, Srivastava’s triple hypergeometric function.

The derivative operator and harmonic number identities

By applying the derivative operator to the corresponding hypergeometric form of a $q$-series transformation due to Andrews [1,Theorem 4], we establish a general harmonic number identity. As the special cases of it, several interesting Chu-Donno type identities and Paule-Schneider type identities are displayed.