Terms Of Hypergeometric Functions Research Articles

Expressing Solutions of Bessel Differential Equation in terms of Hypergeometric Functions

The BDE (Bessel differential equation) is a second-order linear ordinary differential equation (ODE), and it is considered one of the most significant differential equations because of its extensive applications. The solutions of this differential equation are called Bessel functions. These solutions can be expressed in terms of hypergeometric functions, and in this study, the Bessel functions of the first kind are worked. Hypergeometric functions are the sum of a hypergeometric series. A series ∑▒u_n is known as hypergeometric when the ratio u_(n+1)⁄u_n is a rational function of n. BDE has coefficients with variables; therefore, it is solved by the Frobenius approach. We implement some mathematical steps based on the definition of hypergeometric functions to express the solutions in terms of hypergeometric function. The result shows that the solution of the BDE in terms of hypergeometric function is f(x)=a_0 x^p (_1^ )F_2 (1;1+(p+ρ)/2,1+(p-ρ)/2;x^2⁄4) and the two linearly independent solutions for the BDE of order ρ in terms of hypergeometric function are 〖f_1 (x)=a〗_0 x^ρ (_0^ )F_1 (1+ρ;x^2⁄4) and〖〖 f〗_2 (x)=a〗_0 x^ρ (_0^ )F_1 (1-ρ;x^2⁄4).

Fractional electrical impurity

We examine the localized mode and the transmission of plane waves across a capacitive impurity of strength Δ, in a 1D bi-inductive electrical transmission line where the usual discrete Laplacian is replaced by a fractional one characterized by a fractional exponent s. In the absence of the impurity, the plane wave dispersion is computed in closed form in terms of hypergeometric functions. It is observed that the bandwidth decreases steadily, as s decreases towards zero, reaching a minimum width at s = 0. The localized mode energy and spatial profiles are computed in closed form vía lattice Green functions. The profiles show a remnant of the staggered-unstaggered symmetry that is common in non-fractional chains. The width of the localized mode decreases with decreasing s, becoming completely localized at the impurity site at s = 0. The transmission coefficient of plane waves across the impurity is qualitatively similar to its non-fractional counterpart (s = 1), except at low s values ( s≪1 ). For a fixed exponent s, the transmission decreases with increasing Δ.

MultiHypExp: A Mathematica package for expanding multivariate hypergeometric functions in terms of multiple polylogarithms

We present the Mathematica package MultiHypExp that allows for the expansion of multivariate hypergeometric functions (MHFs), especially those likely to appear as solutions of multi-loop, multi-scale Feynman integrals, in the dimensional regularization parameter. The series expansion of MHFs can be carried out around integer values of parameters to express the series coefficients in terms of multiple polylogarithms. The package uses a modified version of the algorithm prescribed in [1]. In the present work, we relate a given MHF to a Taylor series expandable MHF by a differential operator. The Taylor expansion of the latter MHF is found by first finding the associated partial differential equations (PDEs) from its series representation. We then bring the PDEs to the Pfaffian system and further to the canonical form, and solve them order by order in the expansion parameter using appropriate boundary conditions. The Taylor expansion so obtained and the differential operators are used to find the series expansion of the given MHF. We provide examples to demonstrate the algorithm and to describe the usage of the package, which can be found in this url. Program summaryProgram Title:MultiHypExp.wl version 1.0CPC Library link to program files:https://doi.org/10.17632/zxwkx38y7b.1Developer's repository link:https://github.com/souvik5151/MultiHypExpLicensing provisions: GNU General Public License 3Programming language: Wolfram Mathematica version 11.3 or higherExternal routines/libraries: HolonomicFunctions, CANONICA, HYPERDIRE, PolyLogTools, HPLNature of problem: Obtaining the series expansion of multivariate hypergeometric functions around integer values of Pochhammer parameters in terms of multiple polylogarithmsSolution method: Mathematica implementation of a modified version of the algorithm prescribed in Nucl. Phys. B 989 (2023) 116145.

Linear Peridynamics Fourier Multipliers and Eigenvalues

A characterization for the Fourier multipliers and eigenvalues of linear peridynamic operators is provided. The analysis is presented for state-based peridynamic operators for isotropic homogeneous media in any spatial dimension. We provide explicit formulas for the eigenvalues in terms of the space dimension, the nonlocal parameters, and the material properties. The approach we follow is based on the Fourier multiplier analysis developed by Alali and Albin (Applicable Analysis 2526–2546, 1). The Fourier multipliers of linear peridynamic operators are second-order tensor fields, which are given through integral representations. It is shown that the eigenvalues of the peridynamic operators can be derived directly from the eigenvalues of the Fourier multiplier tensors. We reveal a simple structure for the Fourier multipliers in terms of hypergeometric functions, which allows for providing integral representations as well as hypergeometric representations of the eigenvalues. These representations are utilized to show the convergence of the eigenvalues of linear peridynamics to the eigenvalues of the Navier operator of linear elasticity in the limit of vanishing nonlocality. Moreover, the hypergeometric representation of the eigenvalues is utilized to compute the spectrum of linear peridynamic operators.

Integration and calibration of UBCSAND model for drained monotonic and cyclic triaxial compression of aggregates

We report on results from triaxial drained monotonic and cyclic compression tests for four different aggregates (gabbro, limestone, demolished concrete, steel slag), and discuss the applicability of UBCSAND model for simulating the relevant non-linear stress-strain curves. First, UBCSAND is integrated analytically for the elastic and plastic volumetric and deviatoric strains, leading to a pair of novel closed-form solutions expressed in terms of hypergeometric functions. These solutions are inherently free of numerical errors and can be used for the evaluation of results from numerical integration. Second, three calibration techniques for the model parameters are presented and compared: (1) a graphical “chart” solution based on results from forward numerical simulations; (2) a numerical approach based on minimization of residuals; and (3) an informed trial-and-error approach based on the analytical solution. Errors in numerical integration of the model are explored by comparing the analytical solution against results from an Euler integration. Variations in the calibrated parameters are discussed for drained monotonic and cyclic triaxial loading considering different types of aggregates.

The QED four-photon amplitudes off-shell: Part 1

The present paper is the first in a series of four where we use the worldline formalism to obtain the QED four-photon amplitude completely off-shell. We present the result explicitly in terms of hypergeometric functions, and derivatives thereof, for both scalar and spinor QED. The formalism allows us to unify the scalar and spinor loop calculations, avoiding the usual breaking up of the amplitude into Feynman diagrams, and to achieve manifest transversality at the integrand level as well as UV finiteness term by term by an optimized version of the integration-by-parts procedure originally introduced by Bern and Kosower for gluon amplitudes. The full permutation symmetry is maintained throughout, and the amplitudes get projected naturally into the basis of five tensors introduced by Costantini et al. in 1971. Since in many applications of the “four-photon box” some of the photons can be taken in the low-energy limit, and the formalism makes it easy to integrate out any such leg, apart from the case of general kinematics (part 4) we also treat the special cases of one (part 3) or two (part 2) photons taken at low energy. In this first part of the series, we summarize the application of the worldline formalism to the N-photon amplitudes and its relation to Feynman diagrams, derive the optimized tensor-decomposed integrands of the four-photon amplitudes in scalar and spinor QED, and outline the computational strategy to be followed in parts 2 to 4. We also give an overview of the applications of the four-photon amplitudes, with an emphasis on processes that involve some off-shell photons. The case where all photons are taken at low energy (the “Euler-Heisenberg approximation”) is simple enough to be doable for arbitrary photon numbers, and we include it here for completeness.

Generating functions for series involving higher powers of inverse binomial coefficients and their applications

The purpose of this paper is to construct generating functions in terms of hypergeometric function and logarithm function for finite and infinite sums involving higher powers of inverse binomial coefficients. These generating functions provide a novel way of examining higher powers of inverse binomial coefficients from the perspective of these sums, assessing how several of these sums and these coefficients are related to each other. A relation between the Euler–Frobenius polynomial and B‐spline associated with exponential Euler spline is reported. Moreover, with the aid of derivative operator and functional equations for generating functions, many new computational formulas involving the special finite sums of higher powers of (inverse) binomial coefficients, the Bernoulli polynomials and numbers, the Euler polynomials and numbers, the Stirling numbers, the harmonic numbers, and special finite sums are derived. Moreover, a few recurrence relations containing these particular finite sums are given. Using these recurrence relations, we give a solution of the problem which was given by Charalambides. We give calculations algorithms for these finite sums. Applying these algorithms and Wolfram Mathematica 12.0, we give some plots and many values of these polynomials and finite sums.

FeynGKZ: A Mathematica package for solving Feynman integrals using GKZ hypergeometric systems

In the Lee-Pomeransky representation, Feynman integrals can be identified as a subset of Euler-Mellin integrals, which are known to satisfy Gel'fand-Kapranov-Zelevinsky (GKZ) system of partial differential equations. Here we present an automated package to derive the associated GKZ system for a given Feynman diagram and solve it in terms of hypergeometric functions using two equivalent algorithms, namely the triangulation method and the Gröbner deformation method. We present our code in the form of a Mathematica package FeynGKZ.wl which requires the softwares polymake, Macaulay2 and TOPCOM, and the packages AMBRE and Olsson.wl as dependencies. As applications of the package, we find series solutions to the GKZ systems of several one-loop and two-loop Feynman integrals. These are included in the file Examples.nb that can be downloaded along with the package from GitHub. Program summaryProgram Title:FeynGKZ.wl, version 1.0CPC Library link to program files:https://doi.org/10.17632/m8zds756vb.1Developer's repository link:https://github.com/anant-group/FeynGKZLicensing provisions: GNU General Public License 3Programming language:WolframMathematica version 13.0 or higherExternal routines/libraries:Macaulay2 version 1.20, TOPCOM version 0.17.8, polymake version 4.6, AMBRE version 2.1.1 and Olsson.wlNature of problem: Deriving the GKZ system associated with a given Feynman integral, and solving it in terms of multivariate hypergeometric functions.Solution method: Automating the triangulation and Gröbner deformation methods for obtaining Γ-series solutions to the GKZ hypergeometric system associated with a given Feynman integral.

The superradiance phenomenon in spin-one particles

In this paper, we solve the Duffin–Kemmer–Petiau (DKP) equation in the presence of hyperbolic tangent potential for spin-one particles. By partitioning the spin-one spinor, we show that the DKP equation is equivalent to the Klein–Gordon equation formalism. The scattering solutions are derived in terms of hypergeometric functions. The reflection [Formula: see text] and transmission [Formula: see text] coefficients are calculated in terms of the Gamma functions. The results show the presence of the superradiance phenomenon when [Formula: see text] for a specific region in the potential becomes greater than one.

Several families of new exact solutions for second order partial differential equations with variable coefficients

Several families of new exact solutions for a general second order linear partial differential equation with variable coefficients are derived in this paper. All the possible polynomial and polynomial-like solutions of this equation are derived. It is shown that there exist exactly two sets of such families of exact solutions. These solutions are extended to construct different families of exact solutions in terms of hypergeometric functions, which include polynomial solutions as particular cases. A total of eight families of exact solutions are derived using a novel method of balancing powers of the variables simultaneously. Several well known linear partial differential equations in applied mathematics and mechanics are special cases of the general equation considered in this paper and all the polynomial and polynomial-like solutions of these partial differential equations are also explicitly derived as special cases.

Stuckelberg Particle in a Coulomb Field: A Non-Relativistic Approximation

We start with the Stuckelberg tensor system of equations for a boson with spin states S = 1 and S = 0 and fixed intrinsic parity, which is transformed to the matrix form, then generalize this matrix system to the generally covariant case with the use of the tetrad method. This equation is detailed in spherical coordinates in the presence of an external Coulomb field. After separation of the variables we derive the system of 11 radial equations. By diagonalizing the space reflection operator, this system is splitted into two system of four and seven equations for the states with the parities P = (−1) j+1 and P = (−1) j respectively. The system for the states with the parities P = (−1) j+1 leads to the known solution and energy spectrum. The system of seven equations for the states with the parities P = (−1) j is solved for the states with the total angular momentum j = 0 in terms of hypergeometric functions. The system of seven equations for the states with the total angular momenta j = 1, 2, 3, ... turns out to be very complicated, the only nonrelativistic approximation has been studied. The derived nonrelativistic equations are solved in terms of confluent hypergeometric functions, and the corresponding energy spectra are found. In addition, the general form of the nonrelativistic equations for the the Stuckelberg particle is derived in the presence of an arbitrary electromagnetic field.

Some New Results for the Kampé de Fériet Function with an Application

The generalized hypergeometric functions in one and several variables and their natural generalizations appear in many mathematical problems and their applications. The theory of generalized hypergeometric functions in several variables comes from the fact that the solutions of the partial differential equations appearing in a large number of applied problems of mathematical physics have been expressed in terms of such generalized hypergeometric functions. In particular, the Kampé de Fériet function (in two variables) has proved its practical utility in representing solutions to a wide range of problems in pure and applied mathematics, statistics, and mathematical physics. In this context, in a very recent paper, Progri successfully calculated the 2F2 generalized hypergeometric function for a particular set of parameters and expressed the result in terms of the difference between two Kampé de Fériet functions. Inspired by his work, in the present paper, we obtain three results for a terminating 3F2 series of arguments 1 and 2, together with a transformation formula of a 3F2(z) generalized hypergeometric function in terms of the difference between two Kampé de Fériet functions. One application of this result is also provided. The paper concludes with six reduction formulas for the Kampé de Fériet function. Of note, symmetry occurs naturally in the generalized hypergeometric functions pFq and the Kampé de Fériet function involving two variables, which are the two most important functions discussed in this paper.

Exploring the strong-coupling region of SU(N) Seiberg-Witten theory

We consider the Seiberg-Witten solution of pure mathcal{N} = 2 gauge theory in four dimensions, with gauge group SU(N). A simple exact series expansion for the dependence of the 2(N − 1) Seiberg-Witten periods aI(u), aDI(u) on the N − 1 Coulomb-branch moduli un is obtained around the ℤ2N-symmetric point of the Coulomb branch, where all un vanish. This generalizes earlier results for N = 2 in terms of hypergeometric functions, and for N = 3 in terms of Appell functions. Using these and other analytical results, combined with numerical computations, we explore the global structure of the Kähler potential K = frac{1}{2}{sum}_I Im( overline{a} IaDI), which is single valued on the Coulomb branch. Evidence is presented that K is a convex function, with a unique minimum at the ℤ2N-symmetric point. Finally, we explore candidate walls of marginal stability in the vicinity of this point, and their relation to the surface of vanishing Kähler potential.

MHD heat and mass transfer nanofluid flow on a porous cylinder with chemical reaction and viscous dissipation effects: Benchmark solutions

The present study deals with a parametric study of heat and mass transfer of a steady, laminar, and 2D nanofluid flow under the influence of an inclined magnetic field, curvature, permeable medium, thermal radiation, electrical resistance heating, viscous dissipation, heat source/sink, chemical reaction, and concentration power-law exponent effects. Similarity transformations are employed to convert highly nonlinear PDEs to highly nonlinear ODEs. The Frobenius method was used to obtain the heat and mass transfer solutions in terms of hypergeometric function. Iron Oxide Black and water are used as the nanoparticles and base fluid, respectively. Heat and mass transfer boundary conditions are classified as PST (Prescribed surface temperature) and PSC (Prescribed surface concentration), respectively. The effects of nanoparticle volume fraction and curvature parameters on the local skin friction coefficient have been presented. The Box-Behnken design (BBD) is employed to demonstrate the effects of nanoparticle volume fraction parameter, Darcy number, and chemical reaction parameter on the local Sherwood number. Results show that the curvature parameter directly affects the local skin friction coefficient and velocity. In addition, with a rising concentration power-law exponent parameter, the thickness of the concentration boundary layer becomes thinner.

Entropy Analysis in MHD CNTS Flow Due to a Stretching Surface with Thermal Radiation and Heat Source/Sink

The consequence of magnetohydrodynamics (MHD) flow on entropy generation analysis and thermal radiation for carbon nanotubes via a stretched surface through a magnetic field has been discovered. The governing partial differential equations are altered into ordinary differential equations with the aid of the similarity variable. Here, water is considered the base fluid with two types of carbon nanotubes, such as single-wall carbon nanotubes (SWCNTs) and multi-wall carbon nanotubes (MWCNTs). This domain is used in the energy equation, and then it is solved analytically and transferred in terms of hypergeometric function. The existence and nonexistence of solutions for stretching are investigated. Some of the primary findings discussed in this article show that the presence of carbon nanotubes, magnetic field, and Eckert number develop heat transfer in nanofluids and heat sources and that Eckert number reduces entropy formation. Different regulating parameters, such as Casson fluid, mass transpiration, thermal radiation, solid volume fractions, magnetic constraint, and heat source/sink constraint, can be used to analyze the results of velocity and temperature profiles. The novelty of the current study on the influence of magnetic field entropy analysis on CNTs flow with radiation, is that elastic deformation is the subject of this research, and this has not previously been examined. Higher values of heat sources and thermal radiation enhance the heat transfer rate. The study reveals that thermal radiation, Casson fluid; mass transpiration, Darcy number, and Prandtl number increase, and that decrease in the buoyancy ratio, magnetic parameter, and volume fraction decrease the values of the buoyancy ratio, and also control the transfer of heat.

Certain cubic reduction formulas involving hypergeometric functions

In this paper, we obtain a new general double infinite series identity (in terms of the sum of three infinite series) involving the bounded sequence of arbitrary complex numbers using Saalschutz summation theorem for terminating Clausen series. As application of our double series identity, we establish two cubic reduction formulas for Srivastava-Daoust double hypergeometric functions in terms of generalized hypergeometric function with suitable convergence conditions. By the theory of analytic continuation, our cubic reduction formula is also valid in -211/25≤R(z)≤3/4 when F(z)=0, using Mathematica software.

Ising-like and Fibonacci anyons from KZ-equations

In this work we present solutions to Knizhnik-Zamolodchikov (KZ) equations corresponding to conformal block wavefunctions of non-Abelian Ising-like and Fibonacci Anyons. We solve these equations around regular singular points in configuration space in terms of hypergeometric functions and derive explicit monodromy representations of the braid group action. This confirms the correct non-Abelian statistics of the solutions. One novelty of our approach is that we explicitly keep track of spin basis states and identify conformal blocks uniquely with such states at relevant points in moduli space.

Analytical velocity hodograph of projectile motion under polynomial drag

Velocity hodographs are of importance in weather analysis. A historically important case is the hodograph in the Kepler problem. Here, we explore the analytical solutions for the velocity hodograph for the projectile trajectories under a polynomial, in velocity, hydrodynamic dissipation. General solutions in terms of hypergeometric functions are found for odd powers in monomial terms. The even powers have analytical hodographs but we do not find a general expression for them. For completeness we include the constant or Coulomb case, as well as its sum with the linear and quadratic cases. Both shown explicit solutions as functions of the angle. Usually the linear case is discussed in textbooks and articles, here we find that more general cases admit analytical solutions, even when the position and time cannot be obtained explicitly and they need of numerical integration. Furthermore, motion with constant and, additional linear or quadratic drag could be used to design new university lab experiments.

Magnetic and electric black holes in the vector-tensor Horndeski theory

We construct exact solutions of magnetically charged black holes in the vector-tensor Horndeski gravity and discuss their main features. Unlike the analogous electric case, the field equations are linear in a simple (quite standard) parametrization of the metric tensor, and they can be solved analytically even when a cosmological constant is added. The solutions are presented in terms of hypergeometric functions, which makes the analysis of the black hole properties relatively straightforward. Some of the aspects of these black holes are quite ordinary, like the existence of extremal configurations with maximal magnetic charge for a given mass or the existence of a mass with maximal temperature for a given charge, but others are somewhat unexpected, like the existence of black holes with a repulsive gravitational field. We perform our analysis for both signs of the nonminimal coupling constant and find black hole solutions in both cases but with significant differences between them. The most prominent difference is the fact that the black holes for the negative coupling constant have a spherical surface of curvature singularity rather than a single point. On the other hand, the gravitational field produced around this kind of black holes is always attractive. Also, for small enough magnetic charge and negative coupling constant, extremal black holes do not exist, and all magnetic black holes have a single horizon. In addition, we study the trajectories around these magnetic black holes for light as well as massive particles either neutral or electrically charged. Finally, we compare the main features of these black holes with their electric counterparts, adding some aspects that have not been discussed before, like temperature, particle trajectories, and light deflection by electrically charged Horndesky black holes.

Exact scalar (quasi-)normal modes of black holes and solitons in gauged SUGRA

In this paper we identify a new family of black holes and solitons that lead to the exact integration of scalar probes, even in the presence of a non-minimal coupling with the Ricci scalar which has a non-trivial profile. The backgrounds are planar and spherical black holes as well as solitons of SU (2) × SU (2) mathcal{N} = 4 gauged supergravity in four dimensions. On these geometries, we compute the spectrum of (quasi-)normal modes for the non-minimally coupled scalar field. We find that the equation for the radial dependence can be integrated in terms of hypergeometric functions leading to an exact expression for the frequencies. The solutions do not asymptote to a constant curvature spacetime, nevertheless the asymptotic region acquires an extra conformal Killing vector. For the black hole, the scalar probe is purely ingoing at the horizon, and requiring that the solutions lead to an extremum of the action principle we impose a Dirichlet boundary condition at infinity. Surprisingly, the quasinormal modes do not depend on the radius of the black hole, therefore this family of geometries can be interpreted as isospectral in what regards to the wave operator non-minimally coupled to the Ricci scalar. We find both purely damped modes, as well as exponentially growing unstable modes depending on the values of the non-minimal coupling parameter. For the solitons we show that the same integrability property is achieved separately in a non-supersymmetric solutions as well as for the supersymmetric one. Imposing regularity at the origin and a well defined extremum for the action principle we obtain the spectra that can also lead to purely oscillatory modes as well as to unstable scalar probes, depending on the values of the non-minimal coupling.