Hypergeometric Research Articles

Two types of series expansions valid at strong coupling

It is known that perturbative expansions in powers of the coupling in quantum mechanics (QM) and quantum field theory (QFT) are asymptotic series. This can be useful at weak coupling but fails at strong coupling. In this work, we present two types of series expansions valid at strong coupling. We apply the series to a basic integral as well as a QM path integral containing a quadratic and quartic term with coupling constant λ. The first series is the usual asymptotic one, where the quartic interaction is expanded in powers of λ. The second series is an expansion of the quadratic part where the interaction is left alone. This yields an absolutely convergent series in inverse powers of λ valid at strong coupling. For the basic integral, we revisit the first series and identify what makes it diverge even though the original integral is finite. We fix the problem and obtain, remarkably, a series in powers of the coupling which is absolutely convergent and valid at strong coupling. We explain how this series avoids Dyson’s argument on convergence. We then consider the QM path integral (discretized with time interval divided into N equal segments). As before, the second series is absolutely convergent and we obtain analytical expressions in inverse powers of λ for the nth order terms by taking functional derivatives of generalized hypergeometric functions. The expressions are functions of N and we work them out explicitly up to third order. The general procedure has been implemented in a Mathematica program that generates the expressions at any order n. We present numerical results at strong coupling for different values of N starting at N = 2. The series matches the exact numerical value for a given N (up to a certain accuracy). The continuum is formally reached when N → ∞ but in practice this can be reached at small N.

Absolute moments of the variance-gamma distribution

We obtain exact formulas for the absolute raw and central moments of the variance-gamma distribution, as infinite series involving the modified Bessel function of the second kind and the modified Lommel function of the first kind. When the skewness parameter is equal to zero (the symmetric variance-gamma distribution), the infinite series reduces to a single term. Moreover, for the case that the shape parameter is a half-integer (in our parameterisation of the variance-gamma distribution), we obtain a closed-form expression for the absolute moments in terms of confluent hypergeometric functions. As a consequence, we deduce new exact formulas for the absolute raw and central moments of the asymmetric Laplace distribution and the product of two correlated zero mean normal random variables, and more generally the sum of independent copies of such random variables.

Accurate and fast quaternion fractional-order Franklin moments for color image analysis

Quaternion type moments (QTM) based on hypergeometric functions and harmonic functions have attracted interest in the imaging field. The Franklin system is a complete orthonormal system consisting of piecewise linear continuous functions with Haar wavelet collocation points. However, due to the lack of efficient computational methods and explicit expressions, research on piecewise polynomials with Haar wavelet collocation points remains underexplored. Furthermore, the impact of magnitude and phase of quaternion on the color image features extracted by QTM has not been comprehensively explored. To address these problems, a fast and accurate computation method for Franklin functions is proposed. Quaternion fractional-order Franklin moments (QFFM) are proposed and using the Minkowski distance as a prior parameter for selecting unit pure quaternions. Experiments are conducted to demonstrate the superior performance of QFFM in image reconstruction and color image classification tasks compared with QTM-based and deep learning-based methods. Moreover, with the help of experiments, it is confirmed the effectiveness of using the Minkowski distance to select magnitude and phase of quaternion.

Computation of integral transforms in terms of double hypergeometric series ψ2 with applications

The principal aim of the present paper is to evaluate a new integral transform involving the product of two Bessel Functions of the first kind expressed in terms of the Humbert function ψ2. We also derive some interesting special cases from the main result of this paper. The presented result will be applied to generate some new laser beams called as the Humbert beams of type-II and study there propagation behaviour. From graphical simulations this type of wave seems interesting and useful in optical trapping which is used in some applications for pushing and moving particles such as: biological cells, neutral atoms, fluorescent, DNA molecules and trapping productivity on nano-sized spheres.

Existence of quantum states for Klein–Gordon particles based on exact and approximate scenarios

In the present study, Kummer’s eigenvalue spectra from a charged spinless particle located at spherical pseudo-dot of the form r 2 + 1/r 2 is reported. Here, it is shown how confluent hypergeometric functions have principal quantum numbers for considered spatial confinement. To study systematically both constant rest-mass, m 0 c 2 and spatial-varying mass of the radial distribution m 0 c 2 + S(r), the Klein–Gordon equation is solved under exact case and approximate scenario for a constant mass and variable usage, respectively. The findings related to the relativistic eigenvalues of the Klein–Gordon particle moving in the spherical space show the dependence of mass distribution, so it has been obtained that the energy spectra has bigger eigenvalues than m 0 c 2 = 1 fm−1 in exact scenario. Following analysis also shows eigenvalues satisfy the range of E < m 0 c 2 through approximate scenario.

On the matrix versions of the k analog of ℑ‐incomplete Gauss hypergeometric functions and associated fractional calculus

In this paper, our aim is to introduce a new definition of ( ‐incomplete Wright hypergeometric matrix functions ( ‐IWHMFs) using the ‐incomplete Pochhammer matrix symbol. First, we define the ‐incomplete gamma matrix function and introduce the ‐incomplete Pochhammer matrix symbols. Furthermore, we present differential formulas and integral representation related to these ‐IWHMFs. We have also obtained some results regarding the ‐fractional calculus operators of these ‐IWHMFs. Finally, we investigate the solutions of fractional kinetic equations (FKEs) involving the ‐IWHMFs.

Incoherent Detection Performance Analysis of the Distributed Multiple-Input Multiple-Output Radar for Rice Fluctuating Targets

Utilizing spatial diversity, the distributed multiple-input multiple-output (MIMO) radar has the potential advantage of improving system detection performance. In this paper, the incoherent detection performance of distributed multiple-input multiple-output (MIMO) radars is investigated for Rice fluctuating targets. To calculate the incoherent detection probability, the moment generating function (MGF) of the Rice variable is expanded as the infinite series form. By inverting the product of MGFs of multiple independent Rice variables, new closed-form expressions for the probability density function (PDF) of the sum of independent and weighted squares of Rice variables are proposed. The proposed PDF expression for the sum of independent, non-identically distributed (i.n.i.d.) Rice variables involves an infinite series in terms of the confluent Lauricella function. Specially, the PDF for the sum of independent identically distributed (i.i.d.) Rice is expressed as the confluent hypergeometric function-based infinite series. In addition, the uniform convergence of the proposed PDF expression is also validated. Using this proposed expression, the closed-form and approximate expressions of the incoherent detection probability of MIMO radar are derived, respectively. Numerically evaluated results are illustrated and compared with Monte Carlo (MC) simulations to validate the accuracy of the derivations.

Extreme eigenvalues of random matrices from Jacobi ensembles

Two-term asymptotic formulæ for the probability distribution functions for the smallest eigenvalue of the Jacobi β-Ensembles are derived for matrices of large size in the régime where β &amp;gt; 0 is arbitrary and one of the model parameters α1 is an integer. By a straightforward transformation this leads to corresponding results for the distribution of the largest eigenvalue. The explicit expressions are given in terms of multi-variable hypergeometric functions, and it is found that the first-order corrections are proportional to the derivative of the leading order limiting distribution function. In some special cases β = 2 and/or small values of α1, explicit formulæ involving more familiar functions, such as the modified Bessel function of the first kind, are presented.

On Some Domains of Convergence of Branched Continued Fraction Expansions of the Ratios of Horn Hypergeometric Functions H4

On Some Domains of Convergence of Branched Continued Fraction Expansions of the Ratios of Horn Hypergeometric Functions H4

Effect of MHD and radiation on biviscous Bingham fluid flow on Marangoni boundary for heat source/sink with chemical reaction

A significance of Marangoni-driven convective magnetohydrodynamics (MHD) was investigated in a planned study. The present study examines the impact of radiation and chemical reactions on the two dimensional boundary layer flow of bi-viscous Bingham fluid on a thermosolutal Marangoni boundary with magnetic field and heat source/sink. The physical flow problem is mathematically modelled into Navier–Stokes equations. These nonlinear partial differential equations are then mapped into a set of nonlinear ordinary differential equations using similarity transformation. The elegant analytical method is used to obtain analytic approximations for the resulting system of nonlinear differential equations. The analytical solution for the temperature and concentration profiles is expressed in terms of Kummer's confluent hypergeometric function. The features of flow characteristics such as velocity, temperature, and concentration profiles in response to the variations of the emerging parameters are simulated and examined with a physical explanation through graphs. Surface tension is considered to be corresponding to heat and mass. The results reveal that the velocity profile decreases with increasing the magnetic field, while the Marangoni number value increases with increasing the velocity profile. Moreover, the temperature profile rises with rising the radiation parameter and whereas the value chemical reaction parameter upsurges with decline concentration profile. The present problem has great interest due to its applications in industrial applications such as drying of silicon wafers, thin layers of paint, glues, in heat exchangers, crystal growth in space, biological fluids, blood, synovial liquids, plasma, lubricants, many pharmaceutical products, proteins, sewage sludge, china clay, and many more.

P-Adic hypergeometric functions and the trace of Frobenius of elliptic curves

p-Adic hypergeometric functions and the trace of Frobenius of elliptic curves

Extension of Delaunay normalisation for arbitrary powers of the radial distance

In the framework of perturbed Keplerian systems we deal with the Delaunay normalisation of a wide class of perturbations such that the radial distance is raised to an arbitrary real number γ. The averaged function is expressed in terms of the Gauss hypergeometric function 2F1 whereas the associated generating function is the so called Appell hypergeometric function F1. The Gauss hypergeometric function related to the average depends on the eccentricity, e, whereas the Appell function depends additionally on the eccentric anomaly, E, and both special functions are properly defined and evaluated for all e∈[0,1) and E∈[−π,π]. We analyse when the functions we determine can be extended to e=1. When the exponent of the radial distance is an integer, the usual values of the averaged and generating functions are recovered.

Extensions of Bicomplex Hypergeometric Functions and Riemann–Liouville Fractional Calculus in Bicomplex Numbers

In this paper, we present novel advancements in the theory of bicomplex hypergeometric functions and their applications. We extend the hypergeometric function to bicomplex parameters, analyse its convergence region, and define its integral and derivative representations. Furthermore, we delve into the k-Riemann–Liouville fractional integral and derivative within a bicomplex operator, proving several significant theorems. The developed bicomplex hypergeometric functions and bicomplex fractional operators are demonstrated to have practical applications in various fields. This paper also highlights the essential concepts and properties of bicomplex numbers, special functions, and fractional calculus. Our results enhance the overall comprehension and possible applications of bicomplex numbers in mathematical analysis and applied sciences, providing a solid foundation for future research in this field.

Sum representations of Appell–Lauricella functions over finite fields using confluent hypergeometric functions and their applications

Sum representations of Appell–Lauricella functions over finite fields using confluent hypergeometric functions and their applications

Effective secrecy throughput optimization for GTLS-based UWOC systems with eavesdropper outage constraint.

In this work, the effective secrecy throughput (EST) which explicitly captures both the reliability and secrecy features of wiretap channels for an underwater wireless optical communication (UWOC) system has been optimized by considering the generalized transmit laser selection (GTLS) scheme. Specifically, the instantaneous channel state information (CSI) of the main channel and eavesdropper's channel is assumed to be unavailable at the transmitter, which is more practical for the underwater application. The fading of salinity induced oceanic turbulence channels is described by the Weibull distribution considering angular pointing error. The GTLS scheme is first introduced into the UWOC eavesdropping system to improve the quality of the main channel. Based on these models, the analytical closed-form expression of the EST with a constraint on the maximum allowed secrecy outage probability (SOP) is mathematically derived in terms of Meijer's G function and Gauss hypergeometric function. Then, an optimization problem with the objective of maximizing the constrained EST of the proposed UWOC system while ensuring wiretap code rate constraints are formulated. To solve this problem, a whale optimization algorithm (WOA) based wiretap code rates optimization method is effectively developed to acquire the maximum constrained EST, which is validated by the exhaustive search (ES) method. Simulation results suggest that the maximum constrained EST of this UWOC system can be significantly improved by the GTLS scheme. Besides, the maximum allowed SOP has a pronounced impact on the maximum constrained EST. The presented results could offer valuable insights into the physical layer security (PLS) performance enhancement of UWOC systems.

Subclass of Harmonic Univalent Functions Connected with Hypergeometric Functions

In this paper, we introduce and define a new subclass of harmonic univalent functions involving hypergeometric functions which are sense-preserving in the open unit disc. Further we obtain the coefficient bounds, extreme points, convex combination, growth bounds, distortion bounds and integral operator for this class.

Splitting hypergeometric functions over roots of unity

Splitting hypergeometric functions over roots of unity

Padé Approximations and Irrationality Measures on Values of Confluent Hypergeometric Functions

Padé approximations are approximations of holomorphic functions by rational functions. The application of Padé approximations to Diophantine approximations has a long history dating back to Hermite. In this paper, we use the Maier–Chudnovsky construction of Padé-type approximation to study irrationality properties about values of functions with the form f(x)=∑k=0∞xkk!(bk+s)(bk+s+1)⋯(bk+t), where b,t,s are positive integers and obtain upper bounds for irrationality measures of their values at nonzero rational points. Important examples includes exponential integral, Gauss error function and Kummer’s confluent hypergeometric functions.

P-Adic hypergeometric functions and certain weight three newforms

For an odd prime p and a positive integer n, let Gnn[⋯]p denote McCarthy's p-adic hypergeometric function. In this article, we prove p-adic analogue of certain classical hypergeometric identities and using these identities we express the p-th Fourier coefficient of certain weight three newforms in terms of special values of G33[⋯]p. Rodriguez-Villegas conjectured certain supercongruences between values of truncated hypergeometric series and the p-th Fourier coefficients of these newforms. As a consequence of our main results, we obtain another proof of these supercongruences which were earlier proved by Mortenson and Sun.

N-point functions in conformal quantum mechanics: a momentum space odyssey

In this paper, we study the implications of conformal invariance in momentum space for correlation functions in quantum mechanics. We find that three point functions of arbitrary operators can be written in terms of the 2F1 hypergeometric function. We then show that generic four-point functions can be expressed in terms of Appell’s generalized hypergeometric function F2 with one undetermined parameter that plays the role of the conformal cross ratio in momentum space. We also construct momentum space conformal partial waves, which we compare with the Appell F2 representation. We test our expressions against free theory and DFF model correlators, finding an exact agreement. We then analyze five, six, and all higher point functions. We find, quite remarkably, that n-point functions can be expressed in terms of the Lauricella generalized hypergeometric function, EA, with n − 3 undetermined parameters, which is in one-to-one correspondence with the number of conformal cross ratios. This analysis provides the first instance of a closed form for generic momentum space conformal correlators in contrast to the situation in higher dimensions. Further, we show that the existence of multiple solutions to the momentum space Ward identities can be attributed to the Fourier transforms of the various possible time orderings. Finally, we extend our analysis to theories with N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 1, 2 supersymmetry, where we find that the constraints due to the superconformal ward identities are identical to identities involving hypergeometric functions.