Hypergeometric Research Articles

Weighted sum formula for variants of half multiple zeta values

We prove some weighted sum formulas for half multiple zeta values, half finite multiple zeta values, and half symmetric multiple zeta values. The key point of our proof is Dougall’s identity for the generalized hypergeometric function [Formula: see text]. Similar results for interpolated refined symmetric multiple zeta values and half refined symmetric multiple zeta values are also discussed.

CERTAIN BILINEAR GENERATING RELATIONS FOR THE EXTENDED GAUSS HYPERGEOMETRIC FUNCTION, USING FRACTIONAL CALCULUS

The main aim of the present paper is to apply the extended Riemann Liouville fractional derivative operator for finding some bilinear generating relations for extended Gauss hypergeometric function. Two main results are obtained, which are presented in the form of two theorems.

New q-analogues of Van Hamme’s (F.2) supercongruence and of some related supercongruences

Abstract In terms of a very-well-poised 6 ϕ 5 summation formula, the creative microscoping method recently introduced by Guo and Zudilin, and the Chinese remainder theorem for coprime polynomials, we establish four new q-supercongruences for truncated basic hypergeometric series. One of these results is a new q-analogue of the (F.2) supercongruence of Van Hamme.

On asymptotics for lacunary partition functions

Abstract We give asymptotic analysis of power series associated with lacunary partition functions. New partition theoretic interpretations of some basic hypergeometric series are offered as examples.

Hypergeometric expressions for type I Jacobi–Piñeiro orthogonal polynomials with arbitrary number of weights

For a general number p ≥ 2 p\geq 2 of measures, we provide explicit expressions for the Jacobi–Piñeiro and Laguerre of the first kind multiple orthogonal polynomials of type I, presented in terms of generalized hypergeometric functions.

Algebraicity of hypergeometric functions with arbitrary parameters

Algebraicity of hypergeometric functions with arbitrary parameters

Around the Gauss theorem on the values of Euler’s digamma function at rational points

The paper is devoted to new representations of generating functions for the values of the Riemann zeta function at odd points and for certain related numbers in terms of integrals of trigonometric functions depending on a parameter a a . In particular, new integral representations for the Euler digamma function ψ ( a ) \psi (a) are obtained. The resulting integrals can be calculated in terms of the hypergeometric series 3 F 2 {}_3F_{2} and 4 F 3 {}_4F_{3} for some values of the parameters and z = 1 z=1 . Moreover, if a a is a proper rational fraction, then the integrals in question can be reduced to integrals of R ( sin ⁡ x , cos ⁡ x ) R(\sin x, \cos x) , where R R is a rational function of two variables, and are calculated explicitly. In this case, various analogs of the Gauss theorem on the values of the function ψ ( a ) \psi (a) at rational points (and also yet another proof of that theorem) are obtained.

The Generalized Fox–Wright Function: The Laplace Transform, the Erdélyi–Kober Fractional Integral and Its Role in Fractional Calculus

In this paper, we consider and study in detail the generalized Fox–Wright function Ψ˜qp introduced in our recent work as an extension of the Fox–Wright function Ψqp. This special function can be seen as an important case of the so-called I-functions of Rathie and H¯-functions of Inayat-Hussain, that in turn extend the Fox H-functions and appear to include some Feynman integrals in statistical physics, in polylogarithms, in Riemann Zeta-type functions and in other important mathematical functions. Depending on the parameters, Ψ˜qp is an entire function or is analytic in an open disc with a final radius. We derive its basic properties, such as its order and type, and its images under the Laplace transform and under classical fractional-order integrals. Particular cases of Ψ˜qp are specified, including the Mittag-Leffler and Le Roy-type functions and their multi-index analogues and many other special functions of Fractional Calculus. The corresponding results are illustrated. Finally, we emphasize the role of these new generalized hypergeometric functions as eigenfunctions of operators of new Fractional Calculus with specific I-functions as singular kernels. This paper can be considered as a natural supplement to our previous surveys “Going Next after ‘A Guide to Special Functions in Fractional Calculus’: A Discussion Survey”, and “A Guide to Special Functions of Fractional Calculus”, published recently in this journal.

Universal convexity and range problems of shifted hypergeometric functions

In the present paper, we study the shifted hypergeometric function f ( z ) = z 2 F 1 ( a , b ; c ; z ) f(z)=z_{2}F_{1}(a,b;c;z) for real parameters with 0 > a ≤ b ≤ c 0>a\le b\le c and its variant g ( z ) = z 2 F 2 ( a , b ; c ; z 2 ) g(z)=z_{2}F_{2}(a,b;c;z^2) . Our first purpose is to solve the range problems for f f and g g posed by Ponnusamy and Vuorinen [Rocky Mountain J. Math. 31 (2001), pp. 327–353]. Ruscheweyh, Salinas and Sugawa [Israel J. Math. 171 (2009), pp. 285–304] developed the theory of universal prestarlike functions on the slit domain C ∖ [ 1 , + ∞ ) \mathbb {C}\setminus [1,+\infty ) and showed universal starlikeness of f f under some assumptions on the parameters. However, there has been no systematic study of universal convexity of the shifted hypergeometric functions except for the case b = 1 b=1 . Our second purpose is to show universal convexity of f f under certain conditions on the parameters.

Multi-Dimensional Integral Transform with Fox Function in Kernel in Lebesgue-Type Spaces

This paper is devoted to the study of the multi-dimensional integral transform with the Fox H-function in the kernel in weighted spaces with integrable functions in the domain R+n with positive coordinates. Due to the generality of the Fox H-function, many special integral transforms have the form studied in this paper, including operators with such kernels as generalized hypergeometric functions, classical hypergeometric functions, Bessel and modified Bessel functions and so on. Moreover, most important fractional integral operators, such as the Riemann–Liouville type, are covered by the class under consideration. The mapping properties in Lebesgue-weighted spaces, such as the boundedness, the range and the representations of the considered transformation, are established. In special cases, it is applied to the specific integral transforms mentioned above. We use a modern technique based on the extensive use of the Mellin transform and its properties. Moreover, we generalize our own previous results from the one-dimensional case to the multi-dimensional one. The multi-dimensional case is more complex and needs more delicate techniques.

Results for Analytic Function Associated with Briot–Bouquet Differential Subordinations and Linear Fractional Integral Operators

In this paper, we present a new class of linear fractional differential operators that are based on classical Gaussian hypergeometric functions. Then, we utilize the new operators and the concept of differential subordination to construct a convex set of analytic functions. Moreover, through an examination of a certain operator, we establish several notable results related to differential subordination. In addition, we derive inclusion relation results by employing Briot–Bouquet differential subordinations. We also introduce a perspective study for developing subordination results using Gaussian hypergeometric functions and provide certain properties for further research in complex dynamical systems.

Multiple Integrals Involving Pragathi-Satyanarayana’s I-Function, Generalized Gamma and Generalized Hypergeometric Functions

Multiple Integrals Involving Pragathi-Satyanarayana’s I-Function, Generalized Gamma and Generalized Hypergeometric Functions

Monotonicity Results of Ratios between Normalized Tails of Maclaurin Power Series Expansions of Sine and Cosine

In the paper, the authors establish the monotonicity results of the ratios between normalized tails of the Maclaurin power series expansions of the sine and cosine functions and restate them in terms of the generalized hypergeometric functions.

An analytical study on the influence of magnetic field-dependent viscosity on viscous and ohmic dissipative second-grade fluid boundary layers

This article aims to investigate the magnetohydrodynamic (MHD) boundary layer flow of a second-grade non-Newtonian fluid over a horizontally stretching sheet, considering magnetic field-dependent (MFD) viscosity, as well as viscous and Ohmic dissipations. Analytical solutions for previously unexplored momentum and heat transfer equations involving MFD viscosity are derived. Two boundary conditions, namely Prescribed Surface Temperature (PST) and Prescribed Heat Flux (PHF), are taken into account. Governing dimensional partial differential equations (PDEs) are transformed into non-dimensional ordinary differential equations (ODEs) using similarity transformations. Closed-form analytical solutions for flow are derived, considering stretching velocity, MFD viscosity, second-grade fluid properties, and suction impacts. Heat transfer equations are transformed into Gauss-hypergeometric form, yielding solutions in terms of confluent hypergeometric functions. Analytical expressions for skin friction, local Nusselt number, and non-dimensional wall temperature are derived. A unique solution is obtained for the flow equation. It is found that both second-grade and magnetic viscosity parameters expand the momentum boundary layer while the magnetic parameter reduces thickness. Thermal boundary layer thickness increases with a higher second-grade parameter, magnetic viscosity, and Eckert number. Moreover, analytical solutions are validated against published results for a special case.

Bounds for novel extended beta and hypergeometric functions and related results

We introduce a new unified extension of the integral form of Euler’s beta function with a MacDonald function in the integrand and establish functional upper bounds for it. We use this definition to extend as well the Gaussian and Kummer’s confluent hypergeometric functions, for which we provide bounding inequalities. Moreover, we use our extension of the beta function to define a new probability distribution, for which we establish raw moments and moment inequalities and, as by-products, Turán inequalities for the initially defined extended beta function.

Apparent and emergent dark matter around a Schwarzschild black hole

Inspired by the two different dark matter frameworks that were studied recently: one that arises from the non-local effects of entanglement entropy as emergent gravity (characterized by the parameter ξ(M), and zero-point scale length l), and one from dark energy viewed as a superconducting medium (characterized by η(M), and screening length parameter λG), two black hole solutions spherically surrounded with these dark matter models were derived. The effect of these two frameworks on SMBH was analyzed through the resulting deviations in the null regions and the black hole shadow. In addition, constraints to the parameters ξ and η (at 3σ level) were found using the available EHT data for Sgr. A* and M87*. These constraints allow one to deduce the effective mass M, which causes uncertainties in the measurement. On the other hand, if the effective mass is known, one can also deduce the constants associated with ξ, and η. The former framework also introduces an Appell function, a hypergeometric function of two variables that separately allows the analysis of macroscopic and (hypothetical) microscopic black holes. This first framework was found to decrease the radii of the null regions respective to the Schwarzschild counterpart. The shadow radius, however, behaves reversibly. The result of the numerical analysis for the latter framework revealed increased values for the photonsphere and shadow radii. Remarkably, the study also showed that for SMBHs, the amplifying effects of λG are stronger than the scale length l. Finally, results of constraints, as an example, for the upper bound in ξ for M87* indicated that the effective mass causing the deviation was around 2.4×1020M⊙ given that the observed Milgrom’s constant is 5.4×10−10m/s2.

A Note on Exact Closed-Form Solutions for Lamb’s Problem with an Arbitrary Poisson’s Ratio

Abstract Lamb’s Problem, a cornerstone in theoretical seismology, investigates the displacement of an elastic half-space due to surficial or buried point sources. Although the exact closed-form solution for surface displacements from surficial sources is well established, the solutions for buried sources in the 3D case have remained limited, either confined to integral forms or applicable only for certain Poisson’s ratios. This study addresses this gap by generalizing the existing closed-form solutions to accommodate Poisson’s ratio for buried sources. Our solutions are expressed in terms of elementary algebraic expressions, elliptic integrals, and Gauss hypergeometric functions. The derived solution is validated by numerical tests. With the newly derived solution, we also explore the properties of the leaking mode, which stems from the complex conjugate roots of the Rayleigh equation.

Fractional calculus of modified special functions involving the generalized M-series in their kernels and illustrative examples

In this paper we apply the Riemann–Liouville, Erdelyi–Kober and Caputo fractional operators to the modified beta, modified Gauss hypergeometric and modified confluent hypergeometric functions in which the generalized M-series are included in their kernels. Furthermore, as examples, we obtain solutions of some fractional differential equations involving the above modified special functions.

Linear Combination of Order Statistics Moments from Log-Extended Exponential Geometric Distribution with Applications to Entropy

Moments of order statistics (OSs) characterize the Weibull–geometric and half-logistic families of distributions, of which the extended exponential–geometric (EEG) distribution is a particular case. The EEG distribution is used to create the log-extended exponential–geometric (LEEG) distribution, which is bounded in the unit interval (0, 1). In addition to the generalized Stirling numbers of the first kind, a few years ago, the polylogarithm function and the Lerch transcendent function were used to determine the moments of order statistics of the LEEG distributions. As an application based on the L-moments, we expand the features of the LEEG distribution in this work. In terms of the Gauss hypergeometric function, this work presents the precise equations and recurrence relations for the single moments of OSs from the LEEG distribution. Along with recurrence relations between the expectations of function of two OSs from the LEEG distribution, it also displays the truncated and conditional distribution of the OSs. Additionally, we use the L-moments to estimate the parameters of the LEEG distribution. We further fit the LEEG distribution on three practical data sets from medical and environmental sciences areas. It is seen that the estimated parameters through L-moments of the OSs give a superior fit. We finally determine the correspondence between the entropies and the OSs.

Appell-Lauricella hypergeometric functions over finite fields and algebraic varieties

Appell-Lauricella hypergeometric functions over finite fields and algebraic varieties