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This paper is dedicated to deriving novel formulae for the high‐order derivatives of Chebyshev polynomials of the fifth‐kind. The high‐order derivatives of these polynomials are expressed in terms of their original polynomials. The derived formulae contain certain terminating 4F3(1) hypergeometric functions. We show that the resulting hypergeometric functions can be reduced in the case of the first derivative. As an important application—and based on the derived formulas—a spectral tau algorithm is implemented and analyzed for numerically solving the convection–diffusion equation. The convergence and error analysis of the suggested double expansion is investigated assuming that the solution of the problem is separable. Some illustrative examples are presented to check the applicability and accuracy of our proposed algorithm. [ABSTRACT FROM AUTHOR]

In this paper we propose a de Rham-Witt version of the derived KZ equations and their hypergeometric realizations. [ABSTRACT FROM AUTHOR]

The risk assessments during the COVID‐19 pandemic were primarily based on dose–response models derived from the pooled datasets for infection of animals susceptible to SARS‐CoV. Despite similarities, differences in susceptibility between animals and humans exist for respiratory viruses. The two most commonly used dose–response models for calculating the infection risk of respiratory viruses are the exponential and the Stirling approximated β‐Poisson (BP) models. The modified version of the one‐parameter exponential model or the Wells–Riley model was almost solely used for infection risk assessments during the pandemic. Still, the two‐parameter (α and β) Stirling approximated BP model is often recommended compared to the exponential dose–response model due to its flexibility. However, the Stirling approximation restricts this model to the general rules of β ≫ 1 and α ≪ β, and these conditions are very often violated. To refrain from these requirements, we tested a novel BP model by using the Laplace approximation of the Kummer hypergeometric function instead of the conservative Stirling approximation. The datasets of human respiratory airborne viruses available in the literature for human coronavirus (HCoV‐229E) and human rhinovirus (HRV‐16 and HRV‐39) are used to compare the four dose–response models. Based on goodness‐of‐fit criteria, the exponential model was the best fitting model for the HCoV‐229E (k = 0.054) and for HRV‐39 datasets (k = 1.0), whereas the Laplace approximated BP model followed by the exact and Stirling approximated BP models are preferred for both the HRV‐16 (α = 0.152 and β = 0.021 for Laplace BP) and the HRV‐16 and HRV‐39 pooled datasets (α = 0.2247 and β = 0.0215 for Laplace BP). [ABSTRACT FROM AUTHOR]

In this paper, we study power series with coefficients equal to a product of a generic sequence and an explicitly given function of a positive parameter expressible in terms of the Pochhammer symbols. Four types of such series are treated. We show that logarithmic concavity (convexity) of the generic sequence leads to logarithmic concavity (convexity) of the sum of the series with respect to the argument of the explicitly given function. The logarithmic concavity (convexity) is derived from a stronger property, namely, positivity (negativity) of the power series coefficients of the so-called generalized Turánian. Applications to special functions such as the generalized hypergeometric function and the Fox-Wright function are also discussed. [ABSTRACT FROM AUTHOR]

The Bessel function of the first kind J N k x is expanded in a Fourier–Legendre series, as is the modified Bessel function of the first kind I N k x . The purpose of these expansions in Legendre polynomials was not an attempt to rival established numerical methods for calculating Bessel functions but to provide a form for J N k x useful for analytical work in the area of strong laser fields, where analytical integration over scattering angles is essential. Despite their primary purpose, one can easily truncate the series at 21 terms to provide 33-digit accuracy that matches the IEEE extended precision in some compilers. The analytical theme is furthered by showing that infinite series of like-powered contributors (involving   1 F 2 hypergeometric functions) extracted from the Fourier–Legendre series may be summed, having values that are inverse powers of the eight primes 1 / 2 i 3 j 5 k 7 l 11 m 13 n 17 o 19 p multiplying powers of the coefficient k. [ABSTRACT FROM AUTHOR]

In this work, we propose some unified integral formulas for the Mittag-Leffler confluent hypergeometric function (MLCHF), and our findings are assessed in terms of generalized special functions. Additionally, certain unique cases of confluent hypergeometric function have been corollarily presented. [ABSTRACT FROM AUTHOR]

In this article, we get certain integral representations of the multi-index Wright generalized Bessel function by making use of the extended beta function. This function is presented as a part of the generalized Bessel–Maitland function obtained by taking the extended fractional derivative of the generalized Bessel–Maitland function developed by Özarsalan and Özergin [M. Ali Özarslan and E. Özergin, Some generating relations for extended hypergeometric functions via generalized fractional derivative operator, Math. Comput. Model. 52 2010, 9–10, 1825–1833]. In addition, we demonstrate the exciting connections of the multi-index Wright generalized Bessel function with Laguerre polynomials and Whittaker function. Further, we use the generalized Wright hypergeometric function to calculate the Mellin transform and the inverse of the Mellin transform. [ABSTRACT FROM AUTHOR]

Indefinite integrals of products of generalized hypergeometric functions satisfying first- order differential equations are considered. Necessary and sufficient conditions for the algebraic independence of the set of these integrals and of their values at algebraic points are studied. All algebraic identities arising in this case are found in closed form. [ABSTRACT FROM AUTHOR]

We generalise well-known integrals of Ingham-Siegel and Fisher-Hartwig type over the unitary group U(N) with respect to Haar measure, for finite N and including fixed external matrices. When depending only on the eigenvalues of the unitary matrix, such integrals can be related to a Toeplitz determinant with jump singularities. After introducing fixed deterministic matrices as external sources, the integrals can no longer be solved using Andréiéf's integration formula. Resorting to the character expansion as put forward by Balantekin, we derive explicit determinantal formulae containing Kummer's confluent and Gauß' hypergeometric function. They depend only on the eigenvalues of the deterministic matrices and are analytic in the parameters of the jump singularities. Furthermore, unitary two-matrix integrals of the same type are proposed and solved in the same manner. When making part of the deterministic matrices random and integrating over them, we obtain similar formulae in terms of Pfaffian determinants. This is reminiscent to a unitary group integral found recently by Kanazawa and Kieburg [J. Phys. A: Math. Theor. 51(34), 345202 (2018)]. [ABSTRACT FROM AUTHOR]

This paper is a plea for diagonals and telescopers of rational or algebraic functions using creative telescoping, using a computer algebra experimental mathematics learn-by-examples approach. We show that diagonals of rational functions (and this is also the case with diagonals of algebraic functions) are left-invariant when one performs an infinite set of birational transformations on the rational functions. These invariance results generalize to telescopers. We cast light on the almost systematic property of homomorphism to their adjoint of the telescopers of rational or algebraic functions. We shed some light on the reason why the telescopers, annihilating the diagonals of rational functions of the form P / Q k and 1 / Q , are homomorphic. For telescopers with solutions (periods) corresponding to integration over non-vanishing cycles, we have a slight generalization of this result. We introduce some challenging examples of the generalization of diagonals of rational functions, like diagonals of transcendental functions, yielding simple F 1 2 hypergeometric functions associated with elliptic curves, or the (differentially algebraic) lambda-extension of correlation of the Ising model. [ABSTRACT FROM AUTHOR]

The two-parameter Wright special function is an interesting mathematical object that arises in the theory of the space and time-fractional diffusion equations. Moreover, many other special functions are particular instantiations of the Wright function. The article demonstrates finite representations of the Wright function in terms of sums of generalized hypergeometric functions, which in turn provide connections with the theory of the Gaussian, Airy, Bessel, and Error functions, etc. The main application of the presented results is envisioned in computer algebra for testing numerical algorithms for the evaluation of the Wright function. [ABSTRACT FROM AUTHOR]

In this article, we first define and then propose to systematically study some new subclasses of the class of analytic and bi-concave functions in the open unit disk. For this purpose, we make use of a combination of the binomial series and the confluent hypergeometric function. Among some other properties and results, we derive the estimates on the initial Taylor-Maclaurin coefficients | a 2 | and | a 3 | for functions in these analytic and bi-concave function classes, which are introduced in this paper. We also derive a number of corollaries and consequences of our main results in this paper. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We prove general Dwork-type congruences for constant terms attached to tuples of Laurent polynomials. We apply this result to establishing arithmetic and p -adic analytic properties of functions originating from polynomial solutions modulo $p^s$ of hypergeometric and Knizhnik–Zamolodchikov (KZ) equations, solutions which come as coefficients of master polynomials and whose coefficients are integers. As an application, we show that the simplest example of a p -adic KZ connection has an invariant line subbundle while its complex analog has no nontrivial subbundles due to the irreducibility of its monodromy representation. [ABSTRACT FROM AUTHOR]

This paper addresses certain distributional aspects of a scale-free scatter matrix denoted by R that is stemming from a matrix-variate gamma distribution having a positive definite scale parameter matrix B. Under the assumption that B is a diagonal matrix, a structural representation of the determinant of R is derived; the exact density functions of products and ratios of determinants of matrices possessing such a structure are obtained; a closed form expression is given for the density function of R. Moreover, a novel procedure is utilized to establish that certain functions of the determinant of the sample scatter matrix are asymptotically distributed as chi-square or normal random variables. Then, representations of the density function of R that respectively involve multiple integrals, multiple series and Gauss' hypergeometric function are provided for the general case of a positive definite scale parameter matrix, and an illustrative numerical example is presented. Cutting-edge mathematical techniques have been employed to derive the results. Naturally, they also apply to the conventional sample correlation matrix which is encountered in various multivariate inference contexts. [ABSTRACT FROM AUTHOR]

We review briefly the fundamental solutions to some of the most important partial differential operators. These are very crucial in analysis and partial differential equations (PDEs). Among several applications, these are used, for instance, in studying regularity and growth of solutions. [ABSTRACT FROM AUTHOR]

We present a proof of hypergeometric identities which play a crucial role in the theory of Baxter operators in the Ruijsenaars model. [ABSTRACT FROM AUTHOR]

In this paper we construct certain higher Chow cycles in K 1 of the Jacobian of Fermat curves, generalising a construction of Collino. Then we express the regulator of these elements in terms of the special values of hypergeometric functions. [ABSTRACT FROM AUTHOR]

In the survey Kiryakova: "A Guide to Special Functions in Fractional Calculus" (published in this same journal in 2021) we proposed an overview of this huge class of special functions, including the Fox H-functions, the Fox–Wright generalized hypergeometric functions p Ψ q and a large number of their representatives. Among these, the Mittag-Leffler-type functions are the most popular and frequently used in fractional calculus. Naturally, these also include all "Classical Special Functions" of the class of the Meijer's G- and p F q -functions, orthogonal polynomials and many elementary functions. However, it so happened that almost simultaneously with the appearance of the Mittag-Leffler function, another "fractionalized" variant of the exponential function was introduced by Le Roy, and in recent years, several authors have extended this special function and mentioned its applications. Then, we introduced a general class of so-called (multi-index) Le Roy-type functions, and observed that they fall in an "Extended Class of SF of FC". This includes the I-functions of Rathie and, in particular, the H ¯ -functions of Inayat-Hussain, studied also by Buschman and Srivastava and by other authors. These functions initially arose in the theory of the Feynman integrals in statistical physics, but also include some important special functions that are well known in math, like the polylogarithms, Riemann Zeta functions, some famous polynomials and number sequences, etc. The I- and H ¯ -functions are introduced by Mellin–Barnes-type integral representations involving multi-valued fractional order powers of Γ -functions with a lot of singularities that are branch points. Here, we present briefly some preliminaries on the theory of these functions, and then our ideas and results as to how the considered Le Roy-type functions can be presented in their terms. Next, we also introduce Gelfond–Leontiev generalized operators of differentiation and integration for which the Le Roy-type functions are eigenfunctions. As shown, these "generalized integrations" can be extended as kinds of generalized operators of fractional integration, and are also compositions of "Le Roy type" Erdélyi–Kober integrals. A close analogy appears with the Generalized Fractional Calculus with H- and G-kernel functions, thus leading the way to its further development. Since the theory of the I- and H ¯ -functions still needs clarification of some details, we consider this work as a "Discussion Survey" and also provide a list of open problems. [ABSTRACT FROM AUTHOR]

In the present work, utilizing the known series, new series involving reciprocals of binomial coefficients, alternating positive, and negative binomial coefficients are constructed. Further, several new series of reciprocals of binomial coefficients with two odd terms in the denominator are obtained. In the end, we use these to establish the closed form evaluations of hypergeometric functions for the argument 1/16. [ABSTRACT FROM AUTHOR]

Abstract. We construct new (n + 1)-variable models of irreducible representations of the Lie algebra K5. An n-fold integral transformation is used to obtain a new set of models of K5 in terms of difference-differential operators. These models are further exploited to obtain recurrence relations, generating functions and addition theorems. [ABSTRACT FROM AUTHOR]

A new approach to finding the approximate solution of distributed-order fractional optimal control problems (D-O FOCPs) is proposed. This method is based on Fibonacci wavelets (FWs). We present a new Riemann–Liouville operational matrix for FWs using the hypergeometric function. Using this, an operational matrix of the distributed-order fractional derivative is presented. Implementing the mentioned operational matrix with the help of the Gauss–Legendre numerical integration, the problem converts to a system of algebraic equations. Error analysis is proposed. Finally, the validation of the present technique is checked by solving some numerical examples. [ABSTRACT FROM AUTHOR]

This study investigates an extension of the extended Hurwitz-Lerch zeta function. along with related integral images and derivatives. by extending the extended beta function. Also established is a link between the extended Hurwitz-Lerch zeta function and the Laguerre polynomials. It 5.P f is also demonstrated how to use the enlarged Hurwitz-Lerch zeta function Cv.Xcx; c. a.p.q) to probability distributions. Some (old and new) observations are offered here as specific illustrations of our theories. [ABSTRACT FROM AUTHOR]

In this work, the geometric nature of solutions to two second-order differential equations, z y ′ ′ (z) + a (z) y ′ (z) + b (z) y (z) = 0 and z 2 y ′ ′ (z) + a (z) y ′ (z) + b (z) y (z) = d (z) , is studied. Here, a (z) , b (z) , and d (z) are analytic functions defined on the unit disc. Using differential subordination, we established that the normalized solution F (z) (with F(0) = 1) of above differential equations maps the unit disc to the domain bounded by the leminscate curve 1 + z . We construct several examples by the judicious choice of a (z) , b (z) , and d (z) . The examples include Bessel functions, Struve functions, the Bessel–Sturve kernel, confluent hypergeometric functions, and many other special functions. We also established a connection with the nephroid domain. Directly using subordination, we construct functions that are subordinated by a nephroid function. Two open problems are also suggested in the conclusion. [ABSTRACT FROM AUTHOR]

The aim of this work is to establish a connection between Bohr's radius and the analytic and normalized solutions of two differential second-order differential equations, namely y ″ (z) + a (z) y ′ (z) + b (z) y (z) = 0 and z 2 y ″ (z) + a (z) y ′ (z) + b (z) y (z) = d (z) . Using differential subordination, we find the upper bound of the Bohr and Rogosinski radii of the normalized solution F (z) of the above differential equations. We construct several examples by judicious choice of a (z) , b (z) and d (z) . The examples include several special functions like Airy functions, classical and generalized Bessel functions, error functions, confluent hypergeometric functions and associate Laguerre polynomials. [ABSTRACT FROM AUTHOR]

Tsionas [Ann Oper Res 323:229–245, 2023] considered parameter estimation under linex and double-linex loss functions. He was not able to provide details for the double-linex loss function because of not being able to derive a closed form. In this note, we show that a closed form can be derived. Further, we give details of estimation using the double-linex loss function by the method of maximum likelihood and Bayesian method. [ABSTRACT FROM AUTHOR]

To overcome some limits of classical neuronal models, we propose a Markovian generalization of the classical model based on Jacobi processes by introducing downwards jumps to describe the activity of a single neuron. The statistical analysis of interspike intervals is performed by studying the first passage times of the proposed Markovian Jacobi process with jumps through a constant boundary. In particular, we characterize its Laplace transform, which is expressed in terms of some generalization of hypergeometric functions that we introduce, and deduce a closed-form expression for its expectation. Our approach, which is original in the context of first-passage-time problems, relies on intertwining relations between the semigroups of the classical Jacobi process and its generalization, which have been recently established in [P. Cheridito et al., J. Ec. Polytech. - Math., 8 (2021), pp. 331--378]. A numerical investigation of the firing rate of the considered neuron is performed for some choices of the involved parameters and of the jump distributions. [ABSTRACT FROM AUTHOR]

Recently Masjed-Jamei and Koepf obtained generalizations of various classical summation theorems for the 2F1, 3F2, 4F3, 5F4 and 6F5 generalized hypergeometric series. We evaluate a new class of integrals involving generalized hypergeometric function by employing the results given by Masjed-Jamei and Koepf and MacRobert integral, and we give several special cases. [ABSTRACT FROM AUTHOR]

The paper establishes new convergence domains of branched continued fraction expansions of Horn’s hypergeometric function H4 with real and complex parameters. These domains enabled the PC method to establish the analytical extension of analytical functions to their expansions in the studied domains of convergence. A few examples are provided at the end to illustrate this. [ABSTRACT FROM AUTHOR]

In this paper we deduced Turán-type inequalities for nthderivative of k-gamma function, k-gauss hypergeometric functions, kconfluent hypergeometric functions, and k-Appell series F1,k by using different generalizations of Cauchy-Bunyakovsky-Schwarz inequalities with parameter k > 0. Some special cases are also derived. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a q-hypergeometric matrix function rΦs(A,B;Ci;D; ; q; z) and investigate their regions of convergence. We determine some q-matrix contiguous function relations, a q-integral representation and q-difference formulas satisfied by rΦs(A,B;Ci;D; ; q; z) . Certain properties of this matrix function have also been studied from qfractional calculus point of view. Finally, we emphasize on the special cases of rΦs(A,B;Ci;D; ; q; z) [ABSTRACT FROM AUTHOR]

Appell's functions F1-F4 turned out to be particularly useful in solving a variety of problems in both pure and applied mathematics. In literature, there have been published a significant number of interesting and useful results on these functions. In this paper, we prove that the branched continued fraction, which is an expansion of ratio of hypergeometric functions F2converges uniformly to a holomorphic function of two variables on every compact subset of some domain of C²; and that this function is an analytic continuation of such ratio in this domain. As a special case of our main result, we give the representation of hypergeometric functions F2 by a branched continued fraction. To illustrate this, we have given some numerical experiments at the end. [ABSTRACT FROM AUTHOR]

In this paper, we consider some numerical aspects of branched continued fractions as special families of functions to represent and expand analytical functions of several complex variables, including generalizations of hypergeometric functions. The backward recurrence algorithm is one of the basic tools of computation approximants of branched continued fractions. Like most recursive processes, it is susceptible to error growth. Each cycle of the recursive process not only generates its own rounding errors but also inherits the rounding errors committed in all the previous cycles. On the other hand, in general, branched continued fractions are a non-linear object of study (the sum of two fractional-linear mappings is not always a fractional-linear mapping). In this work, we are dealing with a confluent branched continued fraction, which is a continued fraction in its form. The essential difference here is that the approximants of the continued fraction are the so-called figure approximants of the branched continued fraction. An estimate of the relative rounding error, produced by the backward recurrence algorithm in calculating an nth approximant of the branched continued fraction expansion of Horn's hypergeometric function H4, is established. The derivation uses the methods of the theory of branched continued fractions, which are essential in developing convergence criteria. The numerical examples illustrate the numerical stability of the backward recurrence algorithm. [ABSTRACT FROM AUTHOR]

To make a decision to select a suitable approximation for the solution of a functional inequality, we need reliable information. Two useful information ideas are quality and certainty, and the measure of quality and certainty approximation of the solution of a functional inequality helps us to find the optimum approximation. To measure quality and certainty, we used the idea of the Z-number (Z-N) and we introduced the generalized Z-N (GZ-N) as a diagonal matrix of the form diag(X,Y,X*Y), where X is a fuzzy set time-stamped, Y is the probability distribution function and the third part is the fuzzy-random trace of the first and the second subjects. This kind of diagonal matrix allowed us to define a new model of control functions to stabilize our problem. Using stability analysis, we obtained the most suitable approximation for functional inequalities. [ABSTRACT FROM AUTHOR]

In this paper, we introduce ten new quadruple hypergeometric series. We also obtain their various properties such that integral representations, fractional derivatives, N-fractional connections, operational relations and generating functions. [ABSTRACT FROM AUTHOR]

The primary outcome of this study is the construction of partial solutions for a class of multidimensional partial differential equations with multiple singular coefficients of the second order. We consider the generalized multidimensional second-order Euler-Poisson-Darboux equation. Employing a well-known method, we reduce the generalized Euler-Poisson-Darboux equation to a second-order partial differential equation of the hypergeometric type. The solutions to this second order hypergeometric equation comprise 2n functions that contain the first Lauricella hypergeometric function. The Lauricella function, also known as an n-dimensional series, incorporates three distinct parameters - the Pohhammer polynomials. To study the properties of these particular solutions, we require a decomposition formula expressing the first Lauricell function as the product of simpler hypergeometric functions with fewer variables. Through this study of particular solutions and the determination of singularity order at the origin, we establish the uniqueness of these solutions. Thus, having proved the peculiarity of particular solutions at the origin, it can be argued that the constructed particular solutions are fundamental solutions of the generalized multidimensional second-order Euler-Poisson-Darboux equation. [ABSTRACT FROM AUTHOR]

The aim in this note is to obtain new hypergeometric forms for the functions where b is an arbitrary parameter, in terms of Gauss hypergeometric functions. An application of these results (when b = 1 3) is made to obtain the hypergeometric form of the roots of the cubic equation... This complements the entry in the compendium of Prudnikov et al. on page 472, entry (68) of the table, where only the middle root (either real or purely imaginary) is given in hypergeometric form. [ABSTRACT FROM AUTHOR]

In this paper, we have defined new generalizations of some hypergeometric functions and fractional operators with the help of Fox-Wright function. Then, using each of the generalized fractional operators, we derived linear and bilinear generating function relations for these functions. Finally, we have shown that the newly defined hypergeometric functions and fractional operators can be reduced to functions and operators presented in many studies in the literature by giving special values for their parameters. [ABSTRACT FROM AUTHOR]

Two new extensions of the familiar Bernoulli polynomials are considered by using q-sine, q-cosine, q-hypergeometric and q-exponential functions. We call q-sine and q-cosine hypergeometric Bernoulli polynomials. Then, diverse formulas and properties for these polynomials, such as summation formulas, addition formulas, q-derivative properties, q-integral representations and some correlations are derived. Also, q-sine and q-cosine hypergeometric Bernoulli polynomials with two parameters are introduced and some relations and identities are investigated. Furthermore, some computational values are given by tables, and the beautiful zeros representations of the q-sine hypergeometric Bernoulli polynomials and q-cosine hypergeometric Bernoulli polynomials are showed by the figures. [ABSTRACT FROM AUTHOR]

Based on the generalized extended beta function, we shall introduce and study a new hypergeometric-type extended zeta function together with related integral representations, differential relations, finite sums, and series expansions. Also, we present a relationship between the extended zeta function and the Laguerre polynomials. Our hypergeometric type extended zeta function involves several known zeta functions including the Riemann, Hurwitz, Hurwitz-Lerch, and Barnes zeta functions as particular cases. [ABSTRACT FROM AUTHOR]

In this paper, we introduce three types of generalized special functions: beta, Gauss hypergeometric, and confiuent hypergeometric, all involving two generalized M-series at their kernels. We then give several properties of these functions, such as integral representations, functional relations, summation relations, derivative formulas, transformation formulas, and double Laplace transforms. Furthermore, we obtain solutions of fractional partial differential equations involving these new generalized special functions and then we present graphs of the approximate behavior of the solutions. Also, we introduce a new generalized beta distribution and incomplete beta function. Finally, we establish relationships between the new generalized special functions and other generalized special functions found in the literature. [ABSTRACT FROM AUTHOR]

A subset of cycles comprising a permutation σ in the symmetric group Sn, n ∈ N, is called a divisor of σ. Then the partial sums over divisors with sizes up to un, 0 ≤ u ≤ 1, of values of a nonnegative multiplicative function on a random permutation define a stochastic process with nondecreasing trajectories. When normalized the latter is just a random distribution function supported by the unit interval. We establish that its expectations under various weighted probability measures defined on the subsets of Sn are quasihypergeometric distribution functions. Their limits as n → ∞ cover the class of two-parameter beta distributions. It is shown that, under appropriate conditions, the convergence rate is of the negative power of n order. That opens a new possibility to model the beta distributions using divisors of permutations. [ABSTRACT FROM AUTHOR]

We give a definition of generalized hypergeometric functions over finite fields using modified Gauss sums, which enables us to find clear analogy with classical hypergeometric functions over the complex numbers. We study their fundamental properties and prove summation formulas, transformation formulas and product formulas. An application to zeta functions of K3-surfaces is given. In the appendix, we give an elementary proof of the Davenport–Hasse multiplication formula for Gauss sums. [ABSTRACT FROM AUTHOR]

Quality control is essential for modern business success. The traditional statistical process control (SPC), however, lacks efficacy in current high-variety low-volume industrial practices since the historical reference data in Phase I are usually too scarce to infer the in-control process parameters accurately. To solve this 'small data' challenge, a novel Bayesian process monitoring scheme via transfer learning is proposed to facilitate a cross-product data sharing. In particular, a joint prior distribution is taken to explicitly capture the relatedness between the process data of two similar products, through which the process information can be transferred from one product (source domain) to improve the Bayesian inference for the other product (target domain). The posteriors can be derived analytically in closed forms by using generalised hypergeometric functions, thereby leading to a computationally efficient control chart for the online real-time monitoring in Phase II. A user-specified parameter is also provided to enable a better theoretical understanding of the transferability matter and a free practical control of the transferred information across domains. Extensive numerical simulations and real example studies of an assembly process validate the superiority of our proposed scheme in terms of both the false alarm rate and detection capability. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

For a power-law elliptical body, the drag force in a high-speed rarefied gas flow is calculated based on several local models. By solving the variational problem, the exponent in the generatrix for a minimum drag body of various aspect ratio is determined depending on the ellipticity coefficient. [ABSTRACT FROM AUTHOR]

Sufficient conditions are determined on the parameters such that the generalized and normalized Bessel function of the first kind, which is an elementary transform of the hypergeometric function and other related functions belong to subclasses of starlike and convex functions defined in the unit disk associated with the exponential mapping. Several differential subordination implications are derived for analytic functions involving Bessel function and the operator introduced by Baricz et al. [Differential subordinations involving generalized Bessel functions, Bull. Malays. Math. Sci. Soc. 38(3) (2015), 1255–1280]. These results are obtained by constructing suitable class of admissible functions. Examples involving trigonometric and hyperbolic functions are provided to illustrate the obtained results. [ABSTRACT FROM AUTHOR]

The special matrix functions have received significant attention in many fields, such as theoretical physics, number theory, probability theory, and the theory of group representations. In 2017, Dwivedi and Sahai introduced the generalized hypergeometric matrix function using matrix parameters and proved the convergence on | z | = 1 . Recently, hypergeometric matrix functions and their potential applications have played a major role in mathematical physics and engineering. Motivated by aforesaid works and in order to enrich this flourishing field, we investigate the Euler-type integral representations for the generalized hypergeometric matrix function and determine various transformations in terms of hypergeometric matrix functions. Furthermore, unit and half arguments have been provided for several particular cases. [ABSTRACT FROM AUTHOR]

We study the statistical behavior of quantum entanglement in bipartite systems over fermionic Gaussian states as measured by von Neumann entropy. The formulas of average von Neumann entropy with and without particle number constraints have been recently obtained, whereas the main results of this work are the exact yet explicit formulas of variances for both cases. For the latter case of no particle number constraint, the results resolve a recent conjecture on the corresponding variance. Different than the existing methods in computing variances over other generic state models, proving the results of this work relies on a new simplification framework. The framework consists of a set of new tools in simplifying finite summations of what we refer to as dummy summation and re-summation techniques. As a by-product, the proposed framework leads to various new transformation formulas of hypergeometric functions. [ABSTRACT FROM AUTHOR]