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In the present paper, we study the shifted hypergeometric function f(z)=z_{2}F_{1}(a,b;c;z) for real parameters with 0

I n this work we introduce new subclasses of meromorphic functions in punctured unit disk and analyzes numerous connections and various features of these subclasses by making use of the linear operator that is connected to Gaussian hypergeometric functions. [ABSTRACT FROM AUTHOR]

This paper shows that certain   3 F 4 hypergeometric functions can be expanded in sums of pair products of   2 F 3 functions, which reduce in special cases to   2 F 3 functions expanded in sums of pair products of   1 F 2 functions. This expands the class of hypergeometric functions having summation theorems beyond those expressible as pair-products of generalized Whittaker functions,   2 F 1 functions, and   3 F 2 functions into the realm of   p F q functions where p < q for both the summand and terms in the series. In addition to its intrinsic value, this result has a specific application in calculating the response of the atoms to laser stimulation in the Strong Field Approximation. [ABSTRACT FROM AUTHOR]

In this article, we present a new proof of the partial fraction expansion of the cotangent function using the theory of hypergeometric functions. [ABSTRACT FROM AUTHOR]

In this paper, we present an extension of the Karlsson–Minton summation formula for a generalized hypergeometric function with integral parameter differences. Namely, we extend one single negative difference in Karlsson–Minton formula to a finite number of integral negative differences, some of which will be repeated. Next, we continue our study of the generalized hypergeometric function evaluated at unity and with integral positive differences (IPD hypergeometric function at the unit argument). We obtain a recurrence relation that reduces the IPD hypergeometric function at the unit argument to F 3 4 . Finally, we note that Euler–Pfaff-type transformations are always based on summation formulas for finite hypergeometric functions, and we give a number of examples. [ABSTRACT FROM AUTHOR]

In this paper, we present novel advancements in the theory of bicomplex hypergeometric functions and their applications. We extend the hypergeometric function to bicomplex parameters, analyse its convergence region, and define its integral and derivative representations. Furthermore, we delve into the k-Riemann–Liouville fractional integral and derivative within a bicomplex operator, proving several significant theorems. The developed bicomplex hypergeometric functions and bicomplex fractional operators are demonstrated to have practical applications in various fields. This paper also highlights the essential concepts and properties of bicomplex numbers, special functions, and fractional calculus. Our results enhance the overall comprehension and possible applications of bicomplex numbers in mathematical analysis and applied sciences, providing a solid foundation for future research in this field.

This paper demonstrates several of Ostrowski-type inequalities for fuzzy number functions and investigates their connections with other inequalities. Specifically, employing the Aumann integral and the Kulisch–Miranker order, as well as the inclusion order on the space of real and compact intervals, we establish various Ostrowski-type inequalities for fuzzy-valued mappings ( F · V · M s). Furthermore, by employing diverse orders, we establish connections with the classical versions of Ostrowski-type inequalities. Additionally, we explore new ideas and results rooted in submodular measures, accompanied by examples and applications to illustrate our findings. Moreover, by using special functions, we have provided some applications of Ostrowski-type inequalities. [ABSTRACT FROM AUTHOR]

This document titled "Fractional Calculus and Hypergeometric Functions in Complex Analysis" explores the impact of fractional calculus on various scientific and engineering disciplines. It emphasizes the significance of fractional operators in the study of fractional calculus and their applications in complex analysis research, specifically in the theory of univalent functions. The document also introduces hypergeometric functions and their connection to the theory of univalent functions. It compiles 12 research papers that cover topics such as geometric properties of fractional differential operators, logarithmic-related problems of univalent functions, and the study of generalized bi-subordinate functions. This document serves as a valuable resource for researchers interested in these subjects and their applications in complex analysis. Additionally, it provides a summary of three articles published in the Special Issue on "Fractional Calculus and Hypergeometric Functions in Complex Analysis." The first article explores the use of the Sălăgean q-differential operator for meromorphic multivalent functions, introducing new subclasses of functions. The second article presents three general double-series identities using Whipple transformations for terminating generalized hypergeometric functions, which can be used to derive additional identities. The third article defines a new generalized domain based on the quotient of two analytic functions and investigates the upper bounds of certain coefficients and determinants. The authors anticipate that these findings will inspire further research in the field. [Extracted from the article]

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]

Abstract 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.

In this paper, we present novel advancements in the theory of bicomplex hypergeometric functions and their applications. We extend the hypergeometric function to bicomplex parameters, analyse its convergence region, and define its integral and derivative representations. Furthermore, we delve into the k-Riemann–Liouville fractional integral and derivative within a bicomplex operator, proving several significant theorems. The developed bicomplex hypergeometric functions and bicomplex fractional operators are demonstrated to have practical applications in various fields. This paper also highlights the essential concepts and properties of bicomplex numbers, special functions, and fractional calculus. Our results enhance the overall comprehension and possible applications of bicomplex numbers in mathematical analysis and applied sciences, providing a solid foundation for future research in this field. [ABSTRACT FROM AUTHOR]

This study aims to assess the generalized matrix transform (M-transform) of various incomplete types of special functions named generalized incomplete hypergeometric functions, incomplete H-functions, incomplete H̅ functions, incomplete I-functions, all of which possess a matrix argument. The matrix argument in this case is a real symmetric positive definite matrix of size k × k having k(k+1)/2 variables. Here, we establish the special functions with a matrix argument by extending the existing special functions with a scalar argument. Both scalar and matrix arguments are significant in statistical distribution problems, particularly in scenarios where the null hypothesis is not assumed to be true. Additionally, we derived specific cases by extending the univariate cases. [ABSTRACT FROM AUTHOR]

Abstract We show that all Feynman integrals in two Euclidean dimensions with massless propagators and arbitrary non-integer propagator powers can be expressed in terms of single-valued analogues of Aomoto-Gelfand hypergeometric functions. The latter can themselves be written as bilinears of hypergeometric functions, with coefficients that are intersection numbers in a twisted homology group. As an application, we show that all one-loop integrals in two dimensions with massless propagators can be written in terms of Lauricella F D r $$ {F}_D^{(r)} $$ functions, while the L-loop ladder integrals are related to the generalised hypergeometric L+1 F L functions.

In recent years, fractional kinetic equations (FKEs) involving various special functions have been widely used to describe and solve significant problems in control theory, biology, physics, image processing, engineering, astrophysics, and many others. This current work proposes a new solution to fractional λ − kinetic equations based on generalized degenerate hypergeometric functions (GDHFs), which has the potential to be applied to calculate changes in the chemical composition of stars such as the sun. Furthermore, this expanded form can also help to solve various problems with phenomena in physics, such as fractional statistical mechanics, anomalous diffusion, and fractional quantum mechanics. Moreover, some of the well-known outcomes are just special cases of this class of pathway-type solutions involving GDHFs, with greater accuracy, while providing an easily calculable solution. Additionally, numerical graphs of these analytical solutions, using MATLAB Software (latest version 2023b), are also considered. [ABSTRACT FROM AUTHOR]

In this paper, we give a complete description of the generalized hypergeometric functions, introduced by Faraut and Kor´anyi on the Cartan domain. We establish some Gauss type contiguous relations between these functions on the two Cartan domains of type I2 and type IV4 analogous to the classical relations in the one variable case. [ABSTRACT FROM AUTHOR]

This paper shows that certain 3F4 hypergeometric functions can be expanded in sums of pair products of 2F3 functions, which reduce in special cases to 2F3 functions expanded in sums of pair products of 1F2 functions. This expands the class of hypergeometric functions having summation theorems beyond those expressible as pair-products of generalized Whittaker functions, 2F1 functions, and 3F2 functions into the realm of pFq functions where p

The intertwining relationship between sustainability and discrete probability distributions found its significance in decision-making processes and risk assessment frameworks. Count data modeling and its practical applications have gained attention in numerous research studies. This investigation focused on a particular discrete distribution characterized by a single parameter obtained through the survival discretization method. Statistical attributes of this distribution were accurately explicated using generalized hypergeometric functions. The unveiled characteristics highlighted its suitability for analyzing data displaying "right-skewed" asymmetry and possessing extended "heavy" tails. Its failure rate function effectively addressed scenarios marked by a consistent decrease in rates. Furthermore, it proved to be a valuable tool for probabilistic modeling of over-dispersed data. The study introduced various estimation methods such as maximum product of spacings, Anderson-Darling, right-tail Anderson-Darling, maximum likelihood, least-squares, weighted least-squares, percentile, and Cramer-Von-Mises, offering comprehensive explanations. A ranking simulation study was conducted to evaluate the performance of these estimators, employing ranking techniques to identify the most effective estimator across different sample sizes. Finally, real-world sustainability engineering and medical datasets were analyzed to demonstrate the significance and application of the newly introduced model.