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15 pages, 508 KB

Open AccessTechnical Note

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

by Zhuo-Wei Miao and Jianbo Wang

Remote Sens. 2024, 16(17), 3240; https://doi.org/10.3390/rs16173240 - 1 Sep 2024

Cited by 1 | Viewed by 1644

Abstract

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, [...] Read more.

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. Full article

(This article belongs to the Section Environmental Remote Sensing)

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16 pages, 316 KB

Open AccessArticle

On the Analytic Extension of Lauricella–Saran’s Hypergeometric Function F_K to Symmetric Domains

by Roman Dmytryshyn and Vitaliy Goran

Symmetry 2024, 16(2), 220; https://doi.org/10.3390/sym16020220 - 11 Feb 2024

Cited by 16 | Viewed by 1651

Abstract

In this paper, we consider the representation and extension of the analytic functions of three variables by special families of functions, namely branched continued fractions. In particular, we establish new symmetric domains of the analytical continuation of Lauricella–Saran’s hypergeometric function

F_{K}

with [...] Read more.

In this paper, we consider the representation and extension of the analytic functions of three variables by special families of functions, namely branched continued fractions. In particular, we establish new symmetric domains of the analytical continuation of Lauricella–Saran’s hypergeometric function

F_{K}

with certain conditions on real and complex parameters using their branched continued fraction representations. We use a technique that extends the convergence, which is already known for a small domain, to a larger domain to obtain domains of convergence of branched continued fractions and the PC method to prove that they are also domains of analytical continuation. In addition, we discuss some applicable special cases and vital remarks. Full article

(This article belongs to the Special Issue Symmetry in Geometric Theory of Analytic Functions)

10 pages, 293 KB

Open AccessArticle

On Some Formulas for the Lauricella Function

by Ainur Ryskan and Tuhtasin Ergashev

Mathematics 2023, 11(24), 4978; https://doi.org/10.3390/math11244978 - 16 Dec 2023

Cited by 5 | Viewed by 2432

Abstract

Lauricella, G. in 1893 defined four multidimensional hypergeometric functions

F_{A}

,

F_{B}

,

F_{C}

and

F_{D}

. These functions depended on three variables but were later generalized to many variables. Lauricella’s functions are infinite sums of products of variables [...] Read more.

Lauricella, G. in 1893 defined four multidimensional hypergeometric functions

F_{A}

,

F_{B}

,

F_{C}

and

F_{D}

. These functions depended on three variables but were later generalized to many variables. Lauricella’s functions are infinite sums of products of variables and corresponding parameters, each of them has its own parameters. In the present work for Lauricella’s function

F_{A}^{(n)}

, the limit formulas are established, some expansion formulas are obtained that are used to write recurrence relations, and new integral representations and a number of differentiation formulas are obtained that are used to obtain the finite and infinite sums. In the presentation and proof of the obtained formulas, already known expansions and integral representations of the considered

F_{A}^{(n)}

function, definitions of gamma and beta functions, and the Gaussian hypergeometric function of one variable are used. Full article

(This article belongs to the Section C1: Difference and Differential Equations)

18 pages, 330 KB

Open AccessArticle

On the Analytic Continuation of Lauricella–Saran Hypergeometric Function F_K(a₁,a₂,b₁,b₂;a₁,b₂,c₃;z)

by Tamara Antonova, Roman Dmytryshyn and Vitaliy Goran

Mathematics 2023, 11(21), 4487; https://doi.org/10.3390/math11214487 - 30 Oct 2023

Cited by 23 | Viewed by 2201

Abstract

The paper establishes an analytical extension of two ratios of Lauricella–Saran hypergeometric functions

F_{K}

with some parameter values to the corresponding branched continued fractions in their domain of convergence. The PC method used here is based on the correspondence between a formal [...] Read more.

The paper establishes an analytical extension of two ratios of Lauricella–Saran hypergeometric functions

F_{K}

with some parameter values to the corresponding branched continued fractions in their domain of convergence. The PC method used here is based on the correspondence between a formal triple power series and a branched continued fraction. As additional results, analytical extensions of the Lauricella–Saran hypergeometric functions

F_{K} (a_{1}, a_{2}, 1, b_{2}; a_{1}, b_{2}, c_{3}; z)

and

F_{K} (a_{1}, 1, b_{1}, b_{2}; a_{1}, b_{2}, c_{3}; z)

to the corresponding branched continued fractions were obtained. To illustrate this, we provide some numerical experiments at the end. Full article

(This article belongs to the Special Issue Approximation Theory and Applications)

11 pages, 278 KB

Open AccessArticle

Applications of q-Real Numbers to Triple q-Hypergeometric Functions and q-Horn Functions

by Thomas Ernst

Mathematics 2023, 11(10), 2370; https://doi.org/10.3390/math11102370 - 19 May 2023

Viewed by 1327

Abstract

The purpose of this article is to study how q-real numbers can be used for computations of convergence regions, q-integral representations of certain multiple triple q-Lauricella functions. The corresponding q-difference equations are also given without proof. In the process, [...] Read more.

The purpose of this article is to study how q-real numbers can be used for computations of convergence regions, q-integral representations of certain multiple triple q-Lauricella functions. The corresponding q-difference equations are also given without proof. In the process, we slightly improve Exton’s original formulas. We also survey the current attempts to generalize the above functions to triple and quadruple hypergeometric functions. Finally, we compute some q-analogues of transformation formulas for Horn functions. Full article

(This article belongs to the Section C1: Difference and Differential Equations)

26 pages, 449 KB

Open AccessFeature PaperArticle

A Generic Formula and Some Special Cases for the Kullback–Leibler Divergence between Central Multivariate Cauchy Distributions

by Nizar Bouhlel and David Rousseau

Entropy 2022, 24(6), 838; https://doi.org/10.3390/e24060838 - 17 Jun 2022

Cited by 6 | Viewed by 3935

Abstract

This paper introduces a closed-form expression for the Kullback–Leibler divergence (KLD) between two central multivariate Cauchy distributions (MCDs) which have been recently used in different signal and image processing applications where non-Gaussian models are needed. In this overview, the MCDs are surveyed and [...] Read more.

This paper introduces a closed-form expression for the Kullback–Leibler divergence (KLD) between two central multivariate Cauchy distributions (MCDs) which have been recently used in different signal and image processing applications where non-Gaussian models are needed. In this overview, the MCDs are surveyed and some new results and properties are derived and discussed for the KLD. In addition, the KLD for MCDs is showed to be written as a function of Lauricella D-hypergeometric series

F_{D}^{(p)}

. Finally, a comparison is made between the Monte Carlo sampling method to approximate the KLD and the numerical value of the closed-form expression of the latter. The approximation of the KLD by Monte Carlo sampling method are shown to converge to its theoretical value when the number of samples goes to the infinity. Full article

(This article belongs to the Special Issue Information and Divergence Measures)

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12 pages, 354 KB

Open AccessArticle

Dirichlet Averages of Generalized Mittag-Leffler Type Function

by Dinesh Kumar, Jeta Ram and Junesang Choi

Fractal Fract. 2022, 6(6), 297; https://doi.org/10.3390/fractalfract6060297 - 28 May 2022

Cited by 6 | Viewed by 2405

Abstract

Since Gösta Magus Mittag-Leffler introduced the so-called Mittag-Leffler function in 1903 and studied its features in five subsequent notes, passing the first half of the 20th century during which the majority of scientists remained almost unaware of the function, the Mittag-Leffler function and [...] Read more.

Since Gösta Magus Mittag-Leffler introduced the so-called Mittag-Leffler function in 1903 and studied its features in five subsequent notes, passing the first half of the 20th century during which the majority of scientists remained almost unaware of the function, the Mittag-Leffler function and its various extensions (referred to as Mittag-Leffler type functions) have been researched and applied to a wide range of problems in physics, biology, chemistry, and engineering. In the context of fractional calculus, Mittag-Leffler type functions have been widely studied. Since Carlson established the notion of Dirichlet average and its different variations, these averages have been explored and used in a variety of fields. This paper aims to investigate the Dirichlet and modified Dirichlet averages of the R-function (an extended Mittag-Leffler type function), which are provided in terms of Riemann-Liouville integrals and hypergeometric functions of several variables. Principal findings in this article are (possibly) applicable. This article concludes by addressing an open problem. Full article

(This article belongs to the Special Issue Fractional Calculus Operators and the Mittag-Leffler Function)

14 pages, 310 KB

Open AccessArticle

A Solution of a Boundary Value Problem with Mixed Conditions for a Four-Dimensional Degenerate Elliptic Equation

by Zharasbek Baishemirov, Abdumauvlen Berdyshev and Ainur Ryskan

Mathematics 2022, 10(7), 1094; https://doi.org/10.3390/math10071094 - 28 Mar 2022

Cited by 5 | Viewed by 1943

Abstract

The solvability issues of counterpart Holmgren’s boundary value problem with mixed conditions for a degenerate four-dimensional second-order Gellerstedt equation [...] Read more.

The solvability issues of counterpart Holmgren’s boundary value problem with mixed conditions for a degenerate four-dimensional second-order Gellerstedt equation

H (u) \equiv y^{m} z^{k} t^{l} u_{x x} + x^{n} z^{k} t^{l} u_{y y} + x^{n} y^{m} t^{l} u_{z z} + x^{n} y^{m} z^{k} u_{t t} = 0,

m, n, k, l \equiv c o n s t > 0,

are studied in the finite domain

R_{4}^{+}

, where the values of normal derivatives are set on the piecewise smooth part of the boundary and the values of the desired function are set on the remaining part of the boundary. The main results of the work are the proof of the uniqueness of the considered problem solution by using an energy integral’s method and the construction of the solution of counterpart Holmgren’s boundary value problem in explicit form by means of Green’s function method, containing the hypergeometric Lauricella’s function

F_{A}^{(4)}

. Using the corresponding fundamental solution for the considered generalized Gellerstedt equation of elliptic type, we construct Green’s function. In addition, formulas of differentiation, some adjacent relations, decomposition formulas, and various properties of Lauricella’s hypergeometric functions were used to establish the main results for the aforementioned problem. Full article

(This article belongs to the Section C1: Difference and Differential Equations)

8 pages, 257 KB

Open AccessArticle

Some Unified Integrals for Generalized Mittag-Leffler Functions

by Prakash Singh, Shilpi Jain and Carlo Cattani

Axioms 2021, 10(4), 261; https://doi.org/10.3390/axioms10040261 - 19 Oct 2021

Cited by 5 | Viewed by 2702

Abstract

Here, we ascertain generalized integral formulas concerning the product of the generalized Mittag-Leffler function. These integral formulas are described in the form of the generalized Lauricella series. Some special cases are also presented in terms of the Wright hypergeometric function. Full article

(This article belongs to the Special Issue Special Functions Associated with Fractional Calculus)

15 pages, 822 KB

Open AccessArticle

Gottlieb Polynomials and Their q-Extensions

by Esra ErkuŞ-Duman and Junesang Choi

Mathematics 2021, 9(13), 1499; https://doi.org/10.3390/math9131499 - 26 Jun 2021

Cited by 4 | Viewed by 2137

Abstract

Since Gottlieb introduced and investigated the so-called Gottlieb polynomials in 1938, which are discrete orthogonal polynomials, many researchers have investigated these polynomials from diverse angles. In this paper, we aimed to investigate the q-extensions of these polynomials to provide certain q-generating [...] Read more.

Since Gottlieb introduced and investigated the so-called Gottlieb polynomials in 1938, which are discrete orthogonal polynomials, many researchers have investigated these polynomials from diverse angles. In this paper, we aimed to investigate the q-extensions of these polynomials to provide certain q-generating functions for three sequences associated with a finite power series whose coefficients are products of the known q-extended multivariable and multiparameter Gottlieb polynomials and another non-vanishing multivariable function. Furthermore, numerous possible particular cases of our main identities are considered. Finally, we return to Khan and Asif’s q-Gottlieb polynomials to highlight certain connections with several other known q-polynomials, and provide its q-integral representation. Furthermore, we conclude this paper by disclosing our future investigation plan. Full article

(This article belongs to the Special Issue Higher Transcendental Functions and Their Multi-Disciplinary Applications)

6 pages, 738 KB

Open AccessArticle

Certain Unified Integrals Involving a Multivariate Mittag–Leffler Function

by Shilpi Jain, Ravi P. Agarwal, Praveen Agarwal and Prakash Singh

Axioms 2021, 10(2), 81; https://doi.org/10.3390/axioms10020081 - 2 May 2021

Cited by 19 | Viewed by 3009

Abstract

A remarkably large number of unified integrals involving the Mittag–Leffler function have been presented. Here, with the same technique as Choi and Agarwal, we propose the establishment of two generalized integral formulas involving a multivariate generalized Mittag–Leffler function, which are expressed in terms [...] Read more.

A remarkably large number of unified integrals involving the Mittag–Leffler function have been presented. Here, with the same technique as Choi and Agarwal, we propose the establishment of two generalized integral formulas involving a multivariate generalized Mittag–Leffler function, which are expressed in terms of the generalized Lauricella series due to Srivastava and Daoust. We also present some interesting special cases. Full article

(This article belongs to the Special Issue Special Functions Associated with Fractional Calculus)

37 pages, 470 KB

Open AccessReview

Recent Developments of the Lauricella String Scattering Amplitudes and Their Exact SL(K + 3,C) Symmetry

by Sheng-Hong Lai, Jen-Chi Lee and Yi Yang

Symmetry 2021, 13(3), 454; https://doi.org/10.3390/sym13030454 - 10 Mar 2021

Cited by 8 | Viewed by 2267

Abstract

In this review, we propose a new perspective to demonstrate the Gross conjecture regarding the high-energy symmetry of string theory. We review the construction of the exact string scattering amplitudes (SSAs) of three tachyons and one arbitrary string state, or the Lauricella SSA [...] Read more.

In this review, we propose a new perspective to demonstrate the Gross conjecture regarding the high-energy symmetry of string theory. We review the construction of the exact string scattering amplitudes (SSAs) of three tachyons and one arbitrary string state, or the Lauricella SSA (LSSA), in the

26 D

open bosonic string theory. These LSSAs form an infinite dimensional representation of the

S L (K + 3, C)

group. Moreover, we show that the

S L (K + 3, C)

group can be used to solve all the LSSAs and express them in terms of one amplitude. As an application in the hard scattering limit, the LSSA can be used to directly prove the Gross conjecture, which was previously corrected and proved by the method of the decoupling of zero norm states (ZNS). Finally, the exact LSSA can be used to rederive the recurrence relations of SSA in the Regge scattering limit with associated

S L (5, C)

symmetry and the extended recurrence relations (including the mass and spin dependent string BCJ relations) in the nonrelativistic scattering limit with the associated

S L (4, C)

symmetry discovered recently. Full article

(This article belongs to the Section Physics)

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15 pages, 648 KB

Open AccessArticle

On the Triple Lauricella–Horn–Karlsson q-Hypergeometric Functions

by Thomas Ernst

Axioms 2020, 9(3), 93; https://doi.org/10.3390/axioms9030093 - 31 Jul 2020

Cited by 5 | Viewed by 3033

Abstract

The Horn–Karlsson approach to find convergence regions is applied to find convergence regions for triple q-hypergeometric functions. It turns out that the convergence regions are significantly increased in the q-case; just as for q-Appell and q-Lauricella functions, additions are [...] Read more.

The Horn–Karlsson approach to find convergence regions is applied to find convergence regions for triple q-hypergeometric functions. It turns out that the convergence regions are significantly increased in the q-case; just as for q-Appell and q-Lauricella functions, additions are replaced by Ward q-additions. Mostly referring to Krishna Srivastava 1956, we give q-integral representations for these functions. Full article

(This article belongs to the Special Issue Differential and Difference Equations: A Themed Issue Dedicated to Prof. Hari M. Srivastava on the Occasion of his 80th Birthday)

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11 pages, 279 KB

Open AccessArticle

On the Volterra-Type Fractional Integro-Differential Equations Pertaining to Special Functions

by Yudhveer Singh, Vinod Gill, Jagdev Singh, Devendra Kumar and Kottakkaran Sooppy Nisar

Fractal Fract. 2020, 4(3), 33; https://doi.org/10.3390/fractalfract4030033 - 9 Jul 2020

Cited by 3 | Viewed by 3101

Abstract

In this article, we apply an integral transform-based technique to solve the fractional order Volterra-type integro-differential equation (FVIDE) involving the generalized Lorenzo-Hartely function and generalized Lauricella confluent hypergeometric function in terms of several complex variables in the kernel. We also investigate and introduce [...] Read more.

In this article, we apply an integral transform-based technique to solve the fractional order Volterra-type integro-differential equation (FVIDE) involving the generalized Lorenzo-Hartely function and generalized Lauricella confluent hypergeometric function in terms of several complex variables in the kernel. We also investigate and introduce the Elazki transform of Hilfer-derivative, generalized Lorenzo-Hartely function and generalized Lauricella confluent hypergeometric function. In this article, we have established three results that are present in the form of lemmas, which give us new results on the above mentioned three functions, and by using these results we have derived our main results that are given in the form of theorems. Our main results are very general in nature, which gives us some new and known results as a particular case of results established here. Full article

(This article belongs to the Special Issue New Challenges Arising in Engineering Problems with Fractional and Integer Order)

33 pages, 429 KB

Open AccessEditor’s ChoiceReview

The Generalized Hypergeometric Structure of the Ward Identities of CFT’s in Momentum Space in d > 2

by Claudio Corianò and Matteo Maria Maglio

Axioms 2020, 9(2), 54; https://doi.org/10.3390/axioms9020054 - 14 May 2020

Cited by 6 | Viewed by 3405

Abstract

We review the emergence of hypergeometric structures (of F₄ Appell functions) from the conformal Ward identities (CWIs) in conformal field theories (CFTs) in dimensions d > 2. We illustrate the case of scalar 3- and 4-point functions. 3-point functions are associated to [...] Read more.

We review the emergence of hypergeometric structures (of F₄ Appell functions) from the conformal Ward identities (CWIs) in conformal field theories (CFTs) in dimensions d > 2. We illustrate the case of scalar 3- and 4-point functions. 3-point functions are associated to hypergeometric systems with four independent solutions. For symmetric correlators, they can be expressed in terms of a single 3K integral—functions of quadratic ratios of momenta—which is a parametric integral of three modified Bessel K functions. In the case of scalar 4-point functions, by requiring the correlator to be conformal invariant in coordinate space as well as in some dual variables (i.e., dual conformal invariant), its explicit expression is also given by a 3K integral, or as a linear combination of Appell functions which are now quartic ratios of momenta. Similar expressions have been obtained in the past in the computation of an infinite class of planar ladder (Feynman) diagrams in perturbation theory, which, however, do not share the same (dual conformal/conformal) symmetry of our solutions. We then discuss some hypergeometric functions of 3 variables, which define 8 particular solutions of the CWIs and correspond to Lauricella functions. They can also be combined in terms of 4K integral and appear in an asymptotic description of the scalar 4-point function, in special kinematical limits. Full article

(This article belongs to the Special Issue Geometric Analysis and Mathematical Physics)

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