Hypergeometric Representations Research Articles

Hadronic Light-by-Light Corrections to the Muon Anomalous Magnetic Moment

We review the hadronic light-by-light (HLbL) contribution to the muon anomalous magnetic moment. Upcoming measurements will reduce the experimental uncertainty of this observable by a factor of four; therefore, the theoretical precision must improve accordingly to fully harness such an experimental breakthrough. With regards to the HLbL contribution, this implies a study of the high-energy intermediate states that are neglected in dispersive estimates. We focus on the maximally symmetric high-energy regime and in-quark loop approximation of perturbation theory, following the method of the OPE with background fields proposed by Bijnens et al. in 2019 and 2020. We confirm their results regarding the contributions to the muon g−2. For this, we use an alternative computational method based on a reduction in the full quark loop amplitude, instead of projecting on a supposedly complete system of tensor structures motivated by first principles. Concerning scalar coefficients, mass corrections have been obtained by hypergeometric representations of Mellin–Barnes integrals. By our technique, the completeness of such kinematic singularity/zero-free tensor decomposition of the HLbL amplitude is explicitly checked.

Discrete complementary exponential and sine integral functions

Abstract Discrete analogues of the sine integral and complementary exponential integral functions are investigated. Hypergeometric representation, power series, and Laplace transforms are derived for each. The difficulties in extending these definitions to other common trigonometric integral functions are discussed.

Random matrices associated to Young diagrams

We consider the singular values of certain Young diagram shaped random matrices. For block-shaped random matrices, the empirical distribution of the squares of the singular eigenvalues converges almost surely to a distribution whose moments are a generalization of the Catalan numbers. The limiting distribution is the density of a product of rescaled independent Beta random variables and its Stieltjes–Cauchy transform has a hypergeometric representation. In special cases we recover the Marchenko–Pastur and Dykema–Haagerup measures of square and triangular random matrices, respectively. We find a further factorization of the moments in terms of two complex-valued random variables that generalizes the factorization of the Marchenko–Pastur law as product of independent uniform and arcsine random variables.

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.

Spectral properties related to generalized complementary Romanovski–Routh polynomials

Complementary Romanovski–Routh polynomials play an important role in extracting specific properties of orthogonal polynomials. In this work, a generalized form of the Complementary Romanovski–Routh polynomials (GCRR) that has the Gaussian hypergeometric representation and satisfies a particular type of recurrence called $$R_{II}$$ type three term recurrence relation involving two arbitrary parameters is considered. Self perturbation of GCRR polynomials leading to extracting two different types of $$R_{II}$$ type orthogonal polynomials are identified. Spectral properties of these resultant polynomials in terms of tri-diagonal linear pencil are analyzed. The LU decomposition of these pencil matrices provided interesting properties involving biorthogonality. Interlacing properties between the zeros of the polynomials in the discussion are established.

Bernstein-type bounds for beta distribution

This work obtains sharp closed-form exponential concentration inequalities of Bernstein type for the ubiquitous beta distribution, improving upon sub-Gaussian and sub-gamma bounds previously studied in this context. The proof leverages a novel handy recursion of order 2 for central moments of the beta distribution, obtained from the hypergeometric representations of moments; this recursion is useful for obtaining explicit expressions for central moments and various tail approximations.

Jacobi-type functions defined by fractional Bessel derivatives

ABSTRACT For a class of even weight functions, generalizations of the classical Jacobi and Laguerre polynomials are defined via fractional Rodrigues' type formulas using the Bessel operators in the form . Their properties, including hypergeometric representation, differential recurrence relations and fractional boundary value problem, are investigated.

Symmetry of terminating basic hypergeometric series representations of the Askey–Wilson polynomials

In this paper, we explore the symmetric nature of the terminating basic hypergeometric series representations of the Askey–Wilson polynomials and the corresponding terminating basic hypergeometric transformations that these polynomials satisfy. In particular we identify and classify the set of 4 and 7 equivalence classes of terminating balanced ϕ34 and terminating very-well-poised W78 basic hypergeometric series which are connected with the Askey–Wilson polynomials. We study the inversion properties of these equivalence classes and also identify the connection of both sets of equivalence classes with the symmetric group S6, the symmetry group of the terminating balanced ϕ34. We then use terminating balanced ϕ34 and terminating very-well poised W78 transformations to give a broader interpretation of Watson's q-analog of Whipple's theorem and its converse.

On matrix polynomials L(m,δ,λ)n (x)

The aim of this paper is to introduce matrix polynomials L(M,?,?) n (x) and establish some properties viz, hypergeometric representation, generating matrix relations, integral representations, recurrence relations, summation formulas, series relation, fractional integral and derivative operators.

On the matrix version of extended Struve function and its application on fractional calculus

The main goal of this article is to study the extend Struve and extended modified Struve matrix functions by making use of extended Beta matrix function. In particular, we investigate certain important properties of these extended matrix functions such as integral representation, differentiation formula and hypergeometric representation of these functions. Finally, we obtain some results on the transform and fractional calculus of these extended Struve and extended modified Struve matrix functions.

Associated Conformable Fractional Legendre Polynomials

Along with the work of Abul-Ez et al. [37], we introduce the associated conformable fractional Legendre polynomials (ACFLPs), from which the fractional differential equation of ACFLPs is established. Subsequently, some of interesting properties are derived such as generating function, hypergeometric representation, analytical formula, besides various of recurrence relations. Also, orthogonal properties of ACFLPs are developed in conformable context. We append our study by presenting the shifted ACFLPs and driving some of important properties such as Rodrigues’ type representation formula of fractional order derivative and explicit formula. An interesting compact closed-form expression is derived from the definite integral using a convenient analytical formula for the shifted ACFLPs. This result is easily generalized for integrands involving products of an arbitrary number of shifted associated Legendre polynomials in conformable sense.

Gauss Hypergeometric Representations of the Ferrers Function of the Second Kind

We derive all eighteen Gauss hypergeometric representations for the Ferrers function of the second kind, each with a different argument. They are obtained from the eighteen hypergeometric representations of the associated Legendre function of the second kind by using a limit representation. For the 18 hypergeometric arguments which correspond to these representations, we give geometrical descriptions of the corresponding convergence regions in the complex plane. In addition, we consider a corresponding single sum Fourier expansion for the Ferrers function of the second kind. In four of the eighteen cases, the determination of the Ferrers function of the second kind requires the evaluation of the hypergeometric function separately above and below the branch cut at [1,infty). In order to complete these derivations, we use well-known results to derive expressions for the hypergeometric function above and below its branch cut. Finally we give a detailed review of the 1888 paper by Richard Olbricht who was the first to study hypergeometric representations of Legendre functions.

On the matrix version of extended Bessel functions and its application to matrix differential equations

In this paper, we focus on the extensions of the Bessel matrix function and the modified Bessel matrix function. We first introduce the extended Bessel matrix function and the extended modified Bessel matrix function of the first kind by using the extended Beta matrix function. Then we establish the integral representations, differentiation formula, and hypergeometric representation of such functions. Finally, as an application, we study a kind of second-order matrix differential equations. We prove that the extended modified Bessel matrix function is a particular solution to this kind of differential equations.

Properties and Graphical Representations of the 2-Variable Form of the Simsek Polynomials

This article is written with an objective to introduce the 2-variable Simsek polynomials Yn(x,y;λ,δ) and to investigate some properties of these polynomials. The 2-variable forms of Changhee and Daehee polynomials are also considered. Several important recurrence relations involving the Cauchy, Changhee, Daehee and Simsek numbers are derived. The hypergeometric function representation for an integral involving these polynomials is obtained. The graphical representations of the 2-variable Simsek polynomials are considered for suitable values of the index n and parameters λ and δ. The article is concluded with the derivation of non-linear differential equation and related identity for the Simsek numbers.

A New Subclass of Certain Analytic Univalent Functions Associated with Hypergeometric Functions

The main objective of the present paper is to give with using the linear operator theory and also hypergeometric representations of related functions a new special subclass $\mathcal{TS}_{p}(2^{-r},2^{-1}), r\in \mathbb{ Z }^{+}$ of uniformly convex functions and in addition a suitable subclass of starlike functions with negative Taylor coefficients. Furthermore, the provided trailblazer outcomes in presented study are generalized to certain functions classes with fixed finitely many Taylor coefficients.

Fractional order of Legendre-type matrix polynomials

Recently, special functions of fractional order calculus have had many applications in various areas of mathematical analysis, physics, probability theory, optimization theory, graph theory, control systems, earth sciences, and engineering. Very recently, Zayed et al. (Mathematics 8:136, 2020) introduced the shifted Legendre-type matrix polynomials of arbitrary fractional orders and their various applications utilizing Rodrigues matrix formulas. In this line of research, we use the fractional order of Rodrigues formula to provide further investigation on such Legendre polynomials from a different point of view. Some properties, such as hypergeometric representations, continuation properties, recurrence relations, and differential equations, are derived. Moreover, Laplace’s first integral form and orthogonality are obtained.

Hypergeometric Representations and Differential-Difference Relations for Some Kernels Appearing in Mathematical Physics

We investigate the analytic properties of a new class of special functions that appear in the kernels of a class of integral operators underlying the dynamics of matter relaxation processes in attractive fields. These functions, recently introduced by the second author, generate the kernels of the principal parts of these operators and play an important role in understanding their spectral characteristics. We reveal the representations of these functions in terms of the Gauss and Clausen hypergeometric functions and present differential-difference and differential equations they satisfy. Mathematically, the results include calculation of certain trigonometric double integrals and derivation of their other properties. Furthermore, they represent a potentially useful tool in matter relaxation in an external field, the study of nanoelectronic electrolyte-based systems and dynamics of charge carriers in media with obstacles.

Fourier Spectral Methods for Nonlocal Models

Efficient and accurate spectral solvers for nonlocal models in any spatial dimension are presented. The approach we pursue is based on the Fourier multipliers of nonlocal Laplace operators introduced in Alali and Albin (Appl Anal :1–21, 2019). It is demonstrated that the Fourier multipliers, and the eigenvalues in particular, can be computed accurately and efficiently. This is achieved through utilizing the hypergeometric representation of the Fourier multipliers in which their computation in n dimensions reduces to the computation of a 1D smooth function given in terms of 2F3. We use this representation to develop spectral techniques to solve periodic nonlocal time-dependent problems. For linear problems, such as the nonlocal diffusion and nonlocal wave equations, we use the diagonalizability of the nonlocal operators to produce a semi-analytic approach. For nonlinear problems, we present a pseudo-spectral method and apply it to solve a Brusselator model with nonlocal diffusion. Accuracy and efficiency of the spectral solvers are discussed.

Hypergeometric series representations of Feynman integrals by GKZ hypergeometric systems

We show that almost all Feynman integrals as well as their coefficients in a Laurent series in dimensional regularization can be written in terms of Horn hypergeometric functions. By applying the results of Gelfand-Kapranov-Zelevinsky (GKZ) we derive a formula for a class of hypergeometric series representations of Feynman integrals, which can be obtained by triangulations of the Newton polytope ∆G corresponding to the Lee- Pomeransky polynomial G. Those series can be of higher dimension, but converge fast for convenient kinematics, which also allows numerical applications. Further, we discuss possible difficulties which can arise in a practical usage of this approach and give strategies to solve them.

Asymptotics of the generalized Gegenbauer functions of fractional degree

The generalized Gegenbauer functions of fractional degree (GGF-Fs), denoted by rGν(λ)(x) (right GGF-Fs) and lGν(λ)(x) (left GGF-Fs) with x∈(−1,1),λ>−1∕2 and real ν≥0, are special functions (usually non-polynomials), which are defined upon the hypergeometric representation of the classical Gegenbauer polynomial by allowing integer degree to be real fractional degree. Remarkably, the GGF-Fs become indispensable for optimal error estimates of polynomial approximation to singular functions, and have intimate relations with several families of nonstandard basis functions recently introduced for solving fractional differential equations. However, some properties of GGF-Fs, which are important pieces for the analysis and applications, are unknown or under explored. The purposes of this paper are twofold. The first is to show that for λ>−1∕2 and x=cosθ with θ∈(0,π),(sinθ)λrGν(λ)(cosθ)=2λΓ(λ+1∕2)π(ν+λ)λcos((ν+λ)θ−λπ∕2)+Rν(λ)(θ),and derive the precise expression of the “residual” term Rν(λ)(θ) for all real ν≥ν0 (with some ν0>0). With this at our disposal, we obtain the bounds of GGF-Fs uniform in ν. Under an appropriate weight function, the bounds are uniform for θ∈[0,π] as well. Moreover, we can study the asymptotics of GGF-Fs with large fractional degree ν. The second is to present miscellaneous properties of GGF-Fs for better understanding of this family of useful special functions.