Multivariable Hypergeometric Functions Research Articles

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 > 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.

Analytic continuations and numerical evaluation of the Appell F1, F3, Lauricella [formula omitted] and Lauricella-Saran [formula omitted] and their application to Feynman integrals

We present our investigation of the study of two variable hypergeometric series, namely Appell F1 and F3 series, and obtain a comprehensive list of its analytic continuations enough to cover the whole real (x,y) plane, except on their singular loci. We also derive analytic continuations of their 3-variable generalization, the Lauricella FD(3) series and the Lauricella-Saran FS(3) series, leveraging the analytic continuations of F1 and F3, which ensures that the whole real (x,y,z) space is covered, except on the singular loci of these functions. While these studies are motivated by the frequent occurrence of these multivariable hypergeometric functions in Feynman integral evaluation, they can also be used whenever they appear in other branches of mathematical physics. To facilitate their practical use, for analytical and numerical purposes, we provide four packages: AppellF1.wl, AppellF3.wl, LauricellaFD.wl, and LauricellaSaranFS.wl in Mathematica. These packages are applicable for generic as well as non-generic values of parameters, keeping in mind their utilities in the evaluation of the Feynman integrals. We explicitly present various physical applications of these packages in the context of Feynman integral evaluation and compare the results using other packages such as FIESTA. Upon applying the appropriate conventions for numerical evaluation, we find that the results obtained from our packages are consistent. Various Mathematica notebooks demonstrating different numerical results are also provided along with this paper.

Investigation of Integral Transformation Associated with Extended Generalized Srivastava’s Hypergeometric Multi Variable Special Function

In recent year study on multivariate special functions and Integral transformation have been booming. In this work, we have focused on Srivastava hypergeometric function , , and with triple variable. We have discussed the literature study and motivation from the recent works on the extension of Srivastava’s multivariable hypergeometric function , , and . In this paper, the extension of , , and is studied based on the generalized beta function and the generalized Pochhammer’s symbol . Furthermore, the Mellin integral transformation and Inverse Mellin integral transformation have been studied for the based extension of the functions , , and . A few of the most recent uses of these transformations in various scientific and engineering fields are also highlighted in this paper. In general, this work seeks to offer a thorough overview of recent breakthroughs in the importance and applications of several integral transforms of Multivariable functions.

Olsson.wl & ROC2.wl: Mathematica packages for transformations of multivariable hypergeometric functions & regions of convergence for their series representations in the two variables case

We present the Olsson.wlMathematica package which aims to find transformations for some classes of multivariable hypergeometric functions. It is based on a well-known method developed by P. O. M. Olsson [1] (1964) to derive the analytic continuations of the Appell F1 double hypergeometric series using the linear transformations of the Gauss F12 hypergeometric function. We provide a brief description of the method of Olsson and demonstrate the use of the commands of the Olsson.wl package using some examples that are presented in the text and in some ancillary Mathematica notebooks. In particular, we reproduce various results of the literature on multivariable hypergeometric functions and show practical applications of this package in the derivation of novel formulas. In the context of high energy physics, we also demonstrate how it can be used to disentangle some known results about the analytic continuation of some series representations of the one-loop pentagon in multi-Regge kinematics and D=6−2ϵ. We also provide a companion package, called ROC2.wl, which is dedicated to the derivation of the regions of convergence of double hypergeometric series. This package can be used independently of Olsson.wl. Program summaryProgram Title:Olsson.wlCPC Library link to program files:https://doi.org/10.17632/gc63xzwzz5.1Licensing provisions: GNU General Public License v3.0.Programming language: Wolfram Mathematica version 11.3 and beyond.Nature of problem: To find the transformation formulas of multivariable hypergeometric series appearing in Feynman integral calculus.Solution method: Mathematica implementation of the method of Olsson [1]. The method uses the transformation theory of lower variable hypergeometric functions to find the transformation formulas of higher variable hypergeometric functions.Companion package:ROC2.wl, which is a Mathematica package used inside Olsson.wl and can also be used as a standalone package. The package is used to find the region of convergence of double hypergeometric functions, using Horn's theorem.

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.

ϵ-expansion of multivariable hypergeometric functions appearing in Feynman integral calculus

We present a new methodology, suitable for implementation on computer, to perform the $\epsilon$-expansion of hypergeometric functions with linear $\epsilon$ dependent Pochhammer parameters in any number of variables. Our approach allows one to perform Taylor as well as Laurent series expansion of multivariable hypergeometric functions. Each of the coefficients of $\epsilon$ in the series expansion is expressed as a linear combination of multivariable hypergeometric functions with the same domain of convergence as that of the original hypergeometric function. We present illustrative examples of hypergeometric functions in one, two and three variables which are typical of Feynman integral calculus.

Optical Intelligent Reflecting Surface for Mixed Dual-Hop FSO and Beamforming-Based RF System in C-RAN

Optical intelligent reflecting surface (IRS) is an emerging and low-cost technology that can establish a stable communication route in free space optical (FSO) transmission environment with obstacles. In this work, optical IRS-aided dual-hop mixed FSO and RF system is first proposed for cloud radio access network (C-RAN). Specifically, polar codes are introduced to combat turbulence and building sway induced fading in FSO link, and transmit beamforming (TBF) technique is designed to achieve optimal data-rate for RF link. Supposing that the IRS-aided FSO link is subject to exponentiated Weibull distribution with geometric and misalignment loss, whereas the RF link experiences the Fisher-Snedecor <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {F}$ </tex-math></inline-formula> composite fading, the analytical closed-form outage probability expression is derived in terms of Meijer G and Lauricella multivariate hypergeometric functions with decode-and-forward strategy. Exact closed-form average bit error rate expressions are obtained when the relay of C-RAN adopts single antenna and multiantenna beamforming techniques. On the basis of moment generating function of end-to-end signal-to-noise ratio, the ergodic capacity is obtained in terms of univariate and multivariate Fox H-functions over independent but not identically distributed Fisher-Snedecor <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {F}$ </tex-math></inline-formula> fading channels. The correctness of the analytical results is verified through Monte-Carlo simulations. We further provide an asymptotic analysis and discuss the achievable diversity orders.

On the Le Cam Distance Between Multivariate Hypergeometric and Multivariate Normal Experiments

In this short note, we develop a local approximation for the log-ratio of the multivariate hypergeometric probability mass function over the corresponding multinomial probability mass function. In conjunction with the bounds from Carter (2002) and Ouimet (2021) on the total variation between the law of a multinomial vector jittered by a uniform on $(-1/2,1/2)^d$ and the law of the corresponding multivariate normal distribution, the local expansion for the log-ratio is then used to obtain a total variation bound between the law of a multivariate hypergeometric random vector jittered by a uniform on $(-1/2,1/2)^d$ and the law of the corresponding multivariate normal distribution. As a corollary, we find an upper bound on the Le Cam distance between multivariate hypergeometric and multivariate normal experiments.

A New Type Multivariable Multiple Hypergeometric Functions

We define a new type of multivariable multiple hypergeometric functions in this paper, which is inspired by Exton's multiple hypergeometric functions given by in [13]. Then, for these functions, we obtain some certain type linear generating functions. After that, we derive a variety classes of multilinear and multilateral generating functions for a family of the multivariable multiple hypergeometric functions. In addition, by employing the Erkus-Srivastava polynomials (see [11]) and the fourth type multivariable Horn functions (see [13]), we have also provided some of its conclusions.

Hypergeometric Functions at Unit Argument: Simple Derivation of Old and New Identities

The main goal of this paper is to derive a number of identities for the generalized hypergeometric function evaluated at unity and for certain terminating multivariate hypergeometric functions from the symmetries and other properties of Meijer's $G$ function. For instance, we recover two- and three-term Thomae relations for ${}_3F_2$, give two- and three-term transformations for ${}_4F_3$ with one unit shift and ${}_5F_4$ with two unit shifts in the parameters, establish multi-term identities for general ${}_{p}F_{p-1}$ and several transformations for terminating Kamp\'e de F\'eriet and Srivastava $F^{(3)}$ functions. We further present a presumably new formula for analytic continuation of ${}_pF_{p-1}(1)$ in parameters and reveal somewhat unexpected connections between the generalized hypergeometric functions and the generalized and ordinary Bernoulli polynomials. Finally, we exploit some recent duality relations for the generalized hypergeometric and $q$-hypergeometric functions to derive multi-term relations for terminating series.

Q-Derivatives of Multivariable q-Hypergeometric Function with Respect to Their Parameters

We consider the q-derivatives of the Srivastava and Daoust basic multivariable hypergeometric function with respect to the parameters. This function embodies a entire number of various q-hypergeometric series of one and several variables. Explicit equations are given for general case of summation indexes with positive real coefficients. As an example derivatives of q-analog of non-confluent Horn type hypergeometric function $${{H}_{3}}$$ is presented.

Hypergeometric Identities Related to Roberts Reductions of Hyperelliptic Integrals

In this article starting from some reductions of hyperelliptic integrals of genus 3 into elliptic integrals, due to Michael Roberts (A Tract on the addition of Elliptic and hyperelliptic integrals, Hodger, Foster and Co, 1871) we obtain several identities which, to the best of our knowledge, are all new. The strategy used at this purpose is to evaluate Roberts integrals, in two different ways, on one side by means of elliptic integrals, obtained from the Roberts method of reduction and, on the other side, using multivariate hypergeometric functions.

Generalized Holmgren Problem for an Elliptic Equation with Several Singular Coefficients

It has recently been established that all fundamental solutions of a multidimensional singular elliptic equation can be expressed via the well-known multivariate Lauricella hypergeometric function. In the present paper, we prove that the generalized Holmgren problem for an elliptic equation with several singular coefficients has a unique solution and find this solution in closed form. When finding the solution, we use decomposition formulas and some contiguous relationships for the multivariate Lauricella hypergeometric function.

Application of the Biot–Savart law to parabolic vortex segments using elliptic integrals

The Biot-Savart law is used in aerodynamic theory to calculate the velocity induced by curved vortex lines. Explicit formulas are developed, using multivariate Appell hypergeometric functions, for the velocity induced by a general parabolic vortex segment. The formulas are derived by constructing a particular pencil of elliptic curves whose period integrals provide the solution to the induced velocity. We use numerical integration and a perturbation expansion to evaluate the validity of our formulas.

Fundamental Solutions for a Class of Multidimensional Elliptic Equations with Several Singular Coefficients

The main result of the present paper is the construction of fundamental solutions for a class of multidimensional elliptic equations with several singular coefficients. These fundamental solutions are directly connected with multiple hypergeometric functions and the decomposition formula is required for their investigation which would express the multivariable hypergeometric function in terms of products of several simpler hypergeometric functions involving fewer variables. In this paper, such a formula is proved instead of a previously existing recurrence formula.The order of singularity and other properties of the fundamental solutions that are necessary for solving boundary value problems for degenerate second-order elliptic equations are determined

Wave functions for quantum integrable particle systems via partial confluences of multivariate hypergeometric functions

Starting from the hyperoctahedral multivariate hypergeometric function of Heckman and Opdam (associated with the BCn root system), we arrive—via partial confluent limits in the sense of Oshima and Shimeno—at solutions of the eigenvalue equations for the quantum Toda chain with one-sided boundary perturbations of Pöschl-Teller type and for the hyperbolic quantum Calogero-Sutherland system in a Morse potential. With the aid of corresponding degenerations of the (bispectral dual) difference equations for the Heckman-Opdam hyperoctahedral hypergeometric function, it is deduced that the eigensolutions in question are holomorphic in the spectral variable.

On conformal blocks, crossing kernels and multi-variable hypergeometric functions

In this note, we present an alternative representation of the conformal block with external scalars in general spacetime dimensions in terms of a finite summation over Appell fourth hypergeometric function F4. We also construct its generalization to the non-local primary exchange operator with continuous spin and its corresponding Mellin representation which are relevant for Lorentzian spacetime. Using these results we apply the Lorentzian inversion formula to compute the so-called crossing kernel in general spacetime dimensions, the resultant expression can be written as a double infinite summation over certain Kampé de Fériet hypergeometric functions with the correct double trace operator singularity structures. We also include some complementary computations in AdS space, demonstrating the orthogonality of conformal blocks and performing the decompositions.

Exact distribution of largest Schmidt eigenvalue for quantum entanglement

We derive exact analytical formulae for the distribution of the largest Schmidt eigenvalue as an explicit piecewise polynomial with rational coefficients and for its moments as rational numbers by using random matrix theory based on the fixed-trace ensemble in order to study the quantum entanglement. The derivation utilizes a new connection between the multivariate hypergeometric functions and the Painlevé systems. The formulae are compared to numerical experiments performed in the coupled-kicked top system to reveal their sensitivity to the type of underlying dynamics, regular or chaos.

Finite summation formulas of Srivastava’s general triple hypergeometric function

This paper continues the study of finite summation formulas of multivariable hypergeometric functions. Earlier, Wang and Chen [Finite summation formulas of double hypergeometric functions, Int. Transf. Spec. Funct. 28(3) (2016) 239–253] established finite summation formulas of double hypergeometric functions by using some well-known summation theorems. We present here finite summation formulas for Srivastava’s general triple hypergeometric function. Certain particular cases leading to finite summation formulas for Lauricella functions and Srivastava’s triple hypergeometric functions are also presented.

Multipoint conformal blocks in the comb channel

Conformal blocks are the building blocks for correlation functions in conformal field theories. The four-point function is the most well-studied case. We consider conformal blocks for n-point correlation functions. For conformal field theories in dimensions d = 1 and d = 2, we use the shadow formalism to compute n-point conformal blocks, for arbitrary n, in a particular channel which we refer to as the comb channel. The result is expressed in terms of a multivariable hypergeometric function, for which we give series, differential, and integral representations. In general dimension d we derive the 5-point conformal block, for external and exchanged scalar operators.