440 Results Found

The main aim of this research paper is to introduce a new extension of the Gauss hypergeometric function and confluent hypergeometric function by using an extended beta function. Some functional relations, summation relations, integral representation...

In this article, we define an extended version of the Pochhammer symbol and then introduce the corresponding extension of the τ-Gauss hypergeometric function. The basic properties of the extended τ-Gauss hypergeometric function, including int...

Since in many particular cases checking directly the conditions from the definitions of starlikeness or convexity of a function can be difficult, in this paper we use the theory of differential subordination and in particular the method of admissible...

The paper considers the generalized hypergeometric function ${}_{3}F_2,$ which is important in various fields of mathematics, physics, and economics. The method is used, according to which the domains of the analytical continuation of the special f...

In the research presented in this paper, confluent hypergeometric function is embedded in the theory of strong differential superordinations. In order to proceed with the study, the form of the confluent hypergeometric function is adapted taking into...

In this paper, we study the k-Kummer hypergeometric function $M_k(a,c;w)$, a generalization of the classical Kummer confluent hypergeometric function, arising as a solution of the k-confluent hypergeometric differential equation. We establish a Kummer-t...

The idea of inequality has been extended from the real plane to the complex plane through the notion of subordination introduced by Professors Miller and Mocanu in two papers published in 1978 and 1981. With this notion came a whole new theory called...

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 confl...

In this paper, the authors briefly review some closed-form formulas of the Gauss hypergeometric function at specific arguments, alternatively prove four of these formulas, newly extend a closed-form formula of the Gauss hypergeometric function at som...

Due to the great success of hypergeometric functions of one variable, a number of hypergeometric functions of two or more variables have been introduced and explored. Among them, the Kampé de Fériet function and its generalizations have...

The beta integral method proved itself as a simple but nonetheless powerful method for generating hypergeometric identities at a fixed argument. In this paper, we propose a generalization by substituting the beta density with a particular type of Mei...

The Pareto–Feller distribution has been widely used across various disciplines to model “heavy-tailed” phenomena, where extreme events such as high incomes or large losses are of interest. In this paper, we present a new bivariate d...

Classical summation theorems for generalized hypergeometric series constitute an important tool in the study of special functions. Motivated by earlier generalizations of these theorems, this paper derives several new summation formulas for generaliz...

The main objective of this paper is to present classical second-order differential subordination knowledge extended in this study to include new results regarding third-order differential subordinations. The focus of this study is on the main problem...

The paper deals with the problem of construction and investigation of branched continued fraction expansions of special functions of several variables. We give some recurrence relations of Horn hypergeometric functions $H_3.$ By these relations the bran...

The fuzzy differential subordination concept was introduced in 2011, generalizing the concept of differential subordination following a recent trend of adapting fuzzy sets theory to other already-established theories. A prolific tool in obtaining new...

Making use of a post-quantum derivative operator, we define two classes of meromorphic analytic functions. For the considered family of functions, we aim to investigate the sharp bounds’ values in the case of the Fekete–Szegö problem. The study of th...

The paper is related to the classical problem of the rational approximation of analytic functions of one or several variables, particulary the issues that arise in the construction and studying of continued fraction expansions and their multidimensio...

The study done for obtaining the original results of this paper involves the fractional integral of the confluent hypergeometric function and presents its new applications for introducing a certain subclass of analytic functions. Conditions for funct...

In their paper published in 1990, Miller and Mocanu have investigated the special function Gaussian hypergeometric function in view of its relation to the theory of analytic functions, stating conditions for this function to be univalent using $a,b,c\in$...

Sanford S. Miller and Petru T. Mocanu’s theory of second-order differential subordinations was extended for the case of third-order differential subordinations by José A. Antonino and Sanford S. Miller in 2011. In this paper, new results...

The paper considers the problem of representation and extension of Appell’s hypergeometric functions by a special family of functions—branched continued fractions. Here, we establish new symmetric domains of the analytical continuation of...

Geometric function theory has extensively explored the geometric characteristics of analytic functions within symmetric domains. This study analyzes the geometric properties of a specific class of analytic functions employing confluent hypergeometric...

The paper deals with the problem of expansion of the ratios of the confluent hypergeometric function of N variables ${\Phi }_D^{\left(N\right)}(a,\overline{b};c;\overline{z})$ into the branched continued fractions (BCF) of the general form with N branches of branching and invest...

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 contin...

The study on fractional integrals of confluent hypergeometric functions provides interesting subordination and superordination results and inspired the idea of using this operator to introduce a new class of analytic functions. Given the class of fun...

The notions of strong differential subordination and its dual, strong differential superordination, have been introduced as extensions of the classical differential subordination and superordination concepts, respectively. The dual theories have deve...

This article defines a new class of meromorphic parabolic starlike functions in the punctured unit disc $D^{*}=\{z\in \mathbb{C}:0<|z|<1\}$ that includes fixed second coefficients of class $\mathcal{A}{}_{s,c}^d\left(\psi ,\tau ,\nu ,\eta \right)$ and the q- hypergeometric functions. For...

Indefinite integrals of products of exponential functions, power functions and generalized hypergeometric functions of some types are considered. Necessary and sufficient conditions are established for the algebraic independence of large sets of such...

This article uses fractional calculus to create novel links between the well-known Mittag-Leffler functions of one, two, three, and four parameters. Hence, this paper studies several new analytical properties using fractional integration and differen...

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...

In recent years, optical systems near the diffraction limit have been widely used in high-end applications. Evidently, an analytical solution of the point spread function (PSF) will help to enhance both understanding and dealing with the imaging proc...

The present paper deals with a new differential operator denoted by $F_{p,t}^{\delta ,n,b,c,m,\beta }$, whose certain properties are deduced by using well-known earlier studies regarding differential inequalities and the Caratheodory function. The new introd...

The present research aims to present new results regarding the fundamental problem of providing sufficient conditions for finding the best subordinant of a third-order differential superordination. A theorem revealing such conditions is first proved...

Hypergeometric functions and their inequalities have found frequent applications in various fields of mathematical sciences. Motivated by the above, we set up certain inequalities including extended type Gauss hypergeometric function and confluent hy...

The literature has an astonishingly large number of integral formulae involving a range of special functions. In this paper, by using three Beta function formulae, we aim to establish three integral formulas whose integrands are products of the gener...

This paper shows that certain ${}_{3}F_4$ hypergeometric functions can be expanded in sums of pair products of ${}_{1}F_2$ functions. In special cases, the ${}_{3}F_4$ hypergeometric functions reduce to ${}_{2}F_3$ functions. Further special cases allow one to reduce the ${}_{2}F_3$ functi...

Studies regarding the two dual notions are conducted in this paper using Riemann–Liouville fractional integral of q-hypergeometric function for obtaining certain fuzzy differential subordinations and superordinations. Fuzzy best dominants and f...

Essentially, whenever a generalized hypergeometric series can be summed in terms of gamma functions, the result will be important as only a few such summation theorems are available in the literature. In this paper, we apply two identities of general...

We establish a new identity for generalized hypergeometric functions and apply it for first- and second-kind Gauss summation formulas to obtain some new summation formulas. The presented identity indeed extends some results of the recent published pa...

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...

There have been numerous investigations on the hypergeometric series ${}_2{F}_1$ and the generalized hypergeometric series ${}_p{F}_q$ such as differential equations, integral representations, analytic continuations, asymptotic expansions, reduction cases, extension...

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...

Functions that are integrals of products of generalized hypergeometric functions of some kind are considered. The conditions for the representability of these functions as a polynomial in hypergeometric ones are found. The necessary and sufficient co...

This paper introduces extensions $H_{4,p}$ and $X_{8,p}$ of Horn’s double hypergeometric function $H_4$ and Exton’s triple hypergeometric function $X_8$, taking into account recent extensions of Euler’s beta function, hypergeometric function, and c...

The choice of the activation functions in neural networks (NN) are of paramount importance in the training process and the performance of NNs. Therefore, the machine learning community has directed its attention to the development of computationally...

Very recently, the incomplete Pochhammer ratios were defined in terms of the incomplete beta function $B_y(x,z)$ . With the help of these incomplete Pochhammer ratios, we introduce new incomplete Gauss, confluent hypergeometric, and App...

The fractional calculus of special functions has significant importance and applications in various fields of science and engineering. Here, we aim to find the fractional integral and differential formulas of the extended hypergeometric-type function...

A sequence of n trials from a finite population with no replacement is described by the hypergeometric distribution as the number of successes. Calculating the likelihood that factory-produced items would be defective is one of the most popular uses...

This paper considers the Horn’s hypergeometric function $H_3$, which is closely related to other hypergeometric functions and has various mathematical or physical applications. The problem of analytical extension of this function is solved using a...