Lauricella Hypergeometric Function Research Articles

Generalizations of the Jacobi identity to the case of the Lauricella function F D ( N )

ABSTRACT The paper considers the issue of constructing differential relations for the Lauricella hypergeometric function F D ( N ) . The formulas found give explicit expressions for the derivative of the product of the function F D ( N ) and some binomials through the product of other binomials and a combination of adjacent Lauricella functions whose parameters differ by 1 or − 1 . The derived differential relations generalize the Jacobi identity and other differential formulas known in the theory of the Gauss hypergeometric function on the multidimensional case. With special parameter values F D ( N ) , the right-hand side of such Jacobi-type formulas are simplified and have the form of a product of binomials and a polynomial. Such formulas can be used to transform a Cauchy-type integral to the form of the Schwartz–Christoffel integral and have an application to the Riemann–Hilbert problem with piecewise constant coefficients.

Analytical study of particle geodesics around a scale-dependent de Sitter black hole

We give a fully analytical description of radial and angular geodesics for massive particles that travel in the spacetime provided by a (3+1)-dimensional scale-dependent black hole in the cosmological background, for which, the quantum corrections are assumed to be small. We show that the equations of motion for radial orbits can be solved by means of Lauricella hypergeometric functions with different numbers of variables. We then classify the angular geodesics and argue that no planetary bound orbits are available. We calculate the epicyclic frequencies of circular orbits at the potential’s maximum and the deflection angle of scattered particles is also calculated. Finally, we resolve the raised Jacobi inversion problem for the angular motion by means of a genus-2 Riemannian theta function, and the possible orbits are derived and discussed.

M-Lauricella hypergeometric functions: integral representations and solutions of fractional differential equations

In this paper, using the modified beta function involving the generalized M-series in its kernel, we described new extensions for the Lauricella hypergeometric functions $F_{A}^{(r)}$, $F_{B}^{(r)}$, $F_{C}^{(r)}$ and $F_{D}^{(r)}$. Furthermore, we obtained various integral representations for the newly defined extended Lauricella hypergeometric functions. Then, we obtained solution of fractional differential equations involving new extensions of Lauricella hypergeometric functions, as examples.

Constructing basises in solution space of the system of equations for the Lauricella Function FD (N)

The paper considers the issue of constructing basises in the solution space of the system of partial differential equations, which is satisfied by the Lauricella hypergeometric function , depending on N complex variables and having complex parameters , b, c. For an arbitrary number N of variables, we have obtained explicit representations for such basis functions in the vicinity of points and in terms of the Horn type hypergeometric series in N variables. For some of these functions we have obtained formulas of analytic continuation. The found continuation formulas are important for calculating the solution of the Riemann – Hilbert problem with piecewise constant coefficients and studying its geometrical meaning. Besides, these formulas are effective for solving the parameters problem for the Schwarz – Christoffel integral and calculating conformal mapping of complex-shaped polygons.

Moduli of quadrilaterals and quasiconformal reflection

We study the interior and exterior moduli of polygonal quadrilaterals. The main result is a formula for a conformal mapping of the upper half plane onto the exterior of a convex polygonal quadrilateral. We prove this by a careful analysis of the Schwarz-Christoffel transformation and obtain the so-called accessory parameters and then the result in terms of the Lauricella hypergeometric function. This result enables us to understand the dissimilarities of the exterior and interior of a convex polygonal quadrilateral. We also give a Mathematica algorithm for the computation. Then we study the special case of an isosceles trapezoidal polygon L and obtain two-sided estimates for the coefficient of quasiconformal reflection over L in terms of special functions and geometric parameters of L.

Connection Problem for an Extension of q-Hypergeometric Systems

We give an example of solutions of the connection problem associated with a certain system of linear $q$-difference equations recently introduced by Park. The result contains a connection formulas of the $q$-Lauricella hypergeometric function $\varphi_{D}$ and those of the $q$-generalized hypergeometric function ${}_{N+1}\varphi_{N}$ as special cases.

LAURICELLA HYPERGEOMETRIC FUNCTIONS, UNIPOTENT FUNDAMENTAL GROUPS OF THE PUNCTURED RIEMANN SPHERE, AND THEIR MOTIVIC COACTIONS

AbstractThe goal of this paper is to raise the possibility that there exists a meaningful theory of ‘motives’ associated with certain hypergeometric integrals, viewed as functions of their parameters. It goes beyond the classical theory of motives, but should be compatible with it. Such a theory would explain a recent and surprising conjecture arising in the context of scattering amplitudes for a motivic Galois group action on Gauss’s ${}_2F_1$ hypergeometric function, which we prove in this paper by direct means. More generally, we consider Lauricella hypergeometric functions and show, on the one hand, how the coefficients in their Taylor expansions can be promoted, via the theory of motivic fundamental groups, to motivic multiple polylogarithms. The latter are periods of ordinary motives and admit an action of the usual motivic Galois group, which we call the local action. On the other hand, we define lifts of the full Lauricella functions as matrix coefficients in a Tannakian category of twisted cohomology, which inherit an action of the corresponding Tannaka group. We call this the global action. We prove that these two actions, local and global, are compatible with each other, even though they are defined in completely different ways. The main technical tool is to prove that metabelian quotients of generalised Drinfeld associators on the punctured Riemann sphere are hypergeometric functions. We also study single-valued versions of these hypergeometric functions, which may be of independent interest.

Performance analysis of non‐orthogonal multiple access assisted cooperative maritime communication system over two‐wave with diffuse power fading

AbstractNon‐orthogonal multiple access (NOMA) and millimeter‐wave (mmWave) communication are two promising technologies to fulfill the high throughput and reliable communication requirements. Maritime communication systems use a generalized fading at near‐sea‐surface channels (for seashore to ship/boat/fishing fleet communication) to provide reliable communication due to atmospheric turbulence. Experimental research on generalized two‐wave diffuse power (TWDP) fading reveal the footprints of mmWave. This article proposes novel exact and asymptotic closed‐form expressions of average symbol error probability (ASEP) of NOMA users for cooperative maritime communication systems over TWDP fading with the different modulations. The end‐to‐end closed‐form expressions are derived in terms of Appell's and Lauricella's hypergeometric functions. Also, the ASEP is shown using the power allocation factor for different modulation schemes. The proposed system provides better transmission reliability using generalized TWDP fading. Further, the outage probability of the system described above is analyzed using the distance, path loss exponent, transmission rate, and Rician factor as performance metrics for both the NOMA users. Effects of fading outage parameters on the system performance are studied. The end‐to‐end exact numerical results have been verified with the Monte‐Carlo simulations that validate with asymptotic results at a high SNR regime.

The Differential Equations of Gravity-free Double Pendulum: Lauricella Hypergeometric Solutions and Their Inversion

This paper solves in closed form the system of ODEs ruling the 2D motion of a gravity free double pendulum (GFDP), not subjected to any force. In such a way its movement is governed by the initial conditions only. The relevant strongly non linear ODEs have been put back to hyperelliptic quadratures which, through the Integral Representation Theorem (IRT), are driven to the Lauricella hypergeometric functions.
 We compute time laws and trajectories of both point masses forming the GFDP in explicit closed form. Suitable sample problems are carried out in order to prove the method effectiveness.

Lauricella's $F_C$ with finite irreducible monodromy group

This paper presents our study of the conditions under which the monodromy group for Lauricella's hypergeometric function $F_C (a,b,c;x)$ is finite irreducible. We provide these conditions in terms of the parameters $a, b, c$. In addition, we discuss the structure of the finite irreducible monodromy group.

Analytic continuation of Lauricella's function F D (N) for variables close to unit near hyperplanes {z j = z l }

For the Lauricella hypergeometric function with an arbitrary number of variables , we construct formulas for analytic continuation into the vicinity of hyperplanes and their intersections providing that all variables are close to unit. Such formulas represent the function near the point as linear combinations of N–multiple hypergeometric series that are solutions of the same system of partial differential equations as . Such series are the N–dimensional analog of the Kummer solutions known for the Gauss equation. The constructed analytical continuation formulas allow one to effectively calculate the function outside the unit polydisk.

The Bergman kernel function for intersections of some cylindrical domains and Lauricella's hypergeometric function

In this paper, we show that Lauricella's hypergeometric function F8 has a close connection with the Bergman kernel for the intersection of two cylindrical domains defined byD(p1,p2,p3):={z∈C3:|z1|2p1+|z2|2p2<1,|z1|2p1+|z3|2p3<1}. We investigate the boundary behavior of the Bergman kernel on the diagonal (z1,0,0). We also compute the explicit form of the Bergman kernel when (p1,p2,p3)=(1,p2,p3) and (p,1,1). As a consequence, we show that D(1,p2,p3) is a Lu Qi-Keng domain. All results can be generalized to the intersection of cylindrical domains in any higher dimension.

Hypergeometric Expression for the Resolvent of the Discrete Laplacian in Low Dimensions

We present an explicit formula for the resolvent of the discrete Laplacian on the square lattice, and compute its asymptotic expansions around thresholds in low dimensions. As a by-product we obtain a closed formula for the fundamental solution to the discrete Laplacian. For the proofs we express the resolvent in a general dimension in terms of the Appell--Lauricella hypergeometric function of type $C$ outside a disk encircling the spectrum. In low dimensions it reduces to a generalized hypergeometric function, for which certain transformation formulas are available for the desired expansions.

Analytic continuation of Lauricella's function F D (N) for large in modulo variables near hyperplanes {z j = z l }

We consider the Lauricella hypergeometric function , depending on variables , and obtain formulas for its analytic continuation into the vicinity of a singular set which is an intersection of the hyperplanes . It is assumed that all N variables are large in modulo. This formulas represent the function outside of the unit polydisk in the form of linear combinations of other N-multiple hypergeometric series that are solutions of the same system of partial differential equations as . The derived hypergeometric series are N-dimensional analogues of the Kummer solutions that are well known in the theory of the classical hypergeometric Gauss equation.

Towards Feynman rules for conformal blocks

We conjecture a simple set of “Feynman rules” for constructing n-point global conformal blocks in any channel in d spacetime dimensions, for external and exchanged scalar operators for arbitrary n and d. The vertex factors are given in terms of Lauricella hypergeometric functions of one, two or three variables, and the Feynman rules furnish an explicit power-series expansion in powers of cross-ratios. These rules are conjectured based on previously known results in the literature, which include four-, five- and six-point examples as well as the n-point comb channel blocks. We prove these rules for all previously known cases, as well as two new ones: the seven-point block in a new topology, and all even-point blocks in the “OPE channel.” The proof relies on holographic methods, notably the Feynman rules for Mellin amplitudes of tree-level AdS diagrams in a scalar effective field theory, and is easily applicable to any particular choice of a conformal block beyond those considered in this paper.

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.

The Neumann and Dirichlet Problems for One Four-Dimensional Degenerate Elliptic Equation

In the present work we study the Neumann and Dirichlet boundary value problems for a four-dimensional (4D) degenerate elliptic equation by means the fundamental solutions, which were constructed earlier. Fundamental solutions are expressed by Lauricella hypergeometric functions. To prove the uniqueness of the solutions to the problems under consideration, the energy-integral method is used. In the course of proving the existence of solutions of the problems, differentiation formulas, decomposition formulas, some adjacent relations formulas and the autotransformation formula of hypergeometric functions are used. To express problem’s solution in explicit form the Gauss–Ostrogradsky formula is used.

SEP of the M-ary $$\theta $$-QAM signals under $$\eta -\mu $$ fading and AWGN noise in a communication system using spatial diversity

In this paper, an upper bound on the symbol error probability (SEP) of the M-ary $$\theta $$-quadrature amplitude modulation scheme, in a channel subject to additive white Gaussian noise (AWGN) and $$\eta -\mu $$ fading, is calculated by means of the union bound. In addition, an exact and closed-form expression for the pairwise error probability (PEP) of the system is derived. The PEP expression obtained is written in terms of the Beta function and the Lauricella hypergeometric function. Another expression for the PEP, written in terms of the Beta function and the Appell hypergeometric function, and two bounds for the PEP, one lower and one upper, are also presented. An exact expression for the optimum rotation angle for quadrature phase shift keying constellation, considering this channel model, is also determined. In addition, an analysis, in terms of PEP, of the modulation diversity system combined with a maximum ratio combining receiver in channels subject to $$\eta -\mu $$ fading and AWGN noise, is performed. PEP and SEP curves, as a function of signal-to-noise ratio, are plotted and corroborated by simulations performed with the Monte Carlo method under different parameters that characterize the channel mathematically. All the expressions determined in this article are closed and new.

Picard–Vessiot groups of Lauricella's hypergeometric systems EC and Calabi–Yau varieties arising integral representations

We study the Zariski closure of the monodromy group Mon of Lauricella's hypergeometric function F C . If the identity component Mon 0 acts irreducibly, then Mon ¯ ∩ SL 2 n ( C ) must be one of classical groups SL 2 n ( C ) , SO 2 n ( C ) and Sp 2 n ( C ) . We also study Calabi–Yau varieties arising from integral representations of F C .

Fractional Calculus, Fractional Differential Equations and Applications

In this paper, we describe two approaches to the definition of fractional derivatives. We investigate the accuracy of the analysis method for solving the fractional order problem. We also give some improvements for the proof of the existence and uniqueness of the solution in fractional differential equations. Treatment of a fractional derivative operator has been made associated with the extended Appell hypergeometric functions of two or three variables and Lauricella hypergeometric function of three variables.