187 Results Found

In 2012, Chu investigated the generalization of classical Watson–Whipple–Dixon summation theorems in the form of analytical formulas. By employing four generalized Watson summation formulas, the objective of this paper is to evaluate a ne...

We consider the Gauss–Manin differential equations for hypergeometric integrals associated with a family of weighted arrangements of hyperplanes moving parallel to themselves. We reduce these equations modulo a prime integer p and construct polynomia...

In the present research, necessary and sufficient conditions for the univalence of the fractional integral of the Gaussian hypergeometric function are obtained by employing specific approaches from differential subordination theory. The first theorem...

In the present paper, we establish two Erdélyi-type integrals for Saran’s hypergeometric function $F_K$, which has applications in specific branches of applied physics and statistics (see below). We employ methods based on the k-dimensional...

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...

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...

This paper is related to notions adapted from fuzzy set theory to the field of complex analysis, namely fuzzy differential subordinations. Using the ideas specific to geometric function theory from the field of complex analysis, fuzzy differential su...

In this manuscript, by using a new identity, we establish some new Maclaurin-type inequalities for functions whose modulus of the first derivatives are $GA$-convex functions via Hadamard fractional integrals.

Given real parameters $a,b,c$ and integer shifts $n_1,n_2,m$, we consider the ratio $R\left(z\right)={}_2{F}_1(a+n_1,b+n_2;c+m;z)/{}_2{F}_1(a,b;c;z)$ of the Gauss hypergeometric functions. We find a formula for $\mathrm{Im}R(x\pm i0)$ with $x>1$ in terms of real hypergeometric polynomial...

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...

In the theory of generalized hypergeometric functions, classical summation theorems for the series ${}_2{F}_1$ , ${}_3{F}_2$ , ${}_4{F}_3$ , ${}_5{F}_4$ and ${}_7{F}_6$ play a key role. Very recently, Masjed-Jamei and...

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...

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...

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...

This paper investigates a mixed Volterra–Fredholm integral equation of the first kind involving a hypergeometric spatial kernel and a continuous Volterra kernel in time. The analysis is formulated in an admissible even spatial space: the unweig...

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...

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...

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...

In this article, we introduce three general double-series identities using Whipple transformations for terminating generalized hypergeometric ${}_4{F}_3$ and ${}_5{F}_4$ functions. Then, by employing the left-sided Riemann–Liouville fractional integral on thes...

Traditionally, the special function theory has many applications in various areas of mathematical physics, economics, statistics, engineering, and many other branches of science. Inspired by certain recent extensions of the k-analogue of gamma, the P...

This paper investigates the composition structures of certain fractional integral operators whose kernels are certain types of generalized hypergeometric functions. It is shown how composition formulas of these operators can be closely related to the...

Through this article, we will discuss a new extension of the incomplete Wright hypergeometric matrix function by using the extended incomplete Pochhammer matrix symbol. First, we give a generalization of the extended incomplete Wright hypergeometric...

In this paper, we revisit the recent result of Luo, Xu, and Raina on an Erdélyi-type integral for Saran’s three-variable hypergeometric function $F_K$. We provide a new proof of this integral and derive an attractive new integral related to...

In his 1984 Ph.D. thesis, J. Greene defined an analogue of the Euler integral transform for finite field hypergeometric series. Here we consider a special family of matrices which arise naturally in the study of this transform and prove a conjecture...

Fractional calculus image formulas involving various special functions are important for evaluation of generalized integrals and to obtain the solution of differential and integral equations. In this paper, the Saigo’s fractional integral opera...

A collection of functions organized according to their indexing based on non-negative integers is grouped by the common factor of fixed integer N. This grouping results in a summation of N series, each consisting of functions partitioned according to...

This paper provides three classes of q-summation formulas in the form of general contiguous extensions of the first q-Kummer summation theorem. Their derivations are presented by using three methods, which are along the lines of the three types of we...

Our investigation is motivated essentially by the demonstrated applications of the basic (or q-) series and basic (or q-) polynomials, especially the basic (or q-) hypergeometric functions and basic (or q-) hypergeometric polynomials, in many diverse...

In a number of problems in applied mathematics, physics (theoretical and mathematical), statistics, and other fields the hypergeometric functions of one and several variables naturally appear. Hypergeometric functions in one and several variables hav...

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...

Recently, the importance of fractional differential equations in the field of applied science has gained more attention not only in mathematics but also in electrodynamics, control systems, economic, physics, geophysics and hydrodynamics. Among the m...

In this paper, motivated by the contributions of several researchers, including (for example) Riemann, Kronecker, Lerch, Ramanujan, and Glaisher on some special integrals known as Mordell’s integrals, we consider the problem of the evaluation o...

In this article, we prove a new Milne-type inequality involving Hadamard fractional integrals for functions with $GA$-convex first derivatives. The limits of the error estimates involve incomplete gamma and confluent hypergeometric functions. The resul...

Investigation of the generalized trigonometric and hyperbolic functions containing two parameters has been a very active research area over the last decade. We believe, however, that their monotonicity and convexity properties with respect to paramet...

We establish two recurrence relations for some Clausen’s hypergeometric functions with unit argument. We solve them to give the explicit formulas. Additionally, we use the moments of Ramanujan’s generalized elliptic integrals to obtain these recurren...

We extend the classical Euler-type integral representations for the Appell functions $F_1$, $F_2$, and $F_3$, to the appropriate Kampé de Fériet functions by using integration against the Meijer–Nørlund G-function. In particular, th...

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 fractional integral is a function known for the elegant results obtained when introducing new operators; it has proved to have interesting applications. In the present paper, differential subordinations and superodinations for the fractional inte...

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...

We state the product formulae of the values of the levels of functions at critical points involved in asymptotic behaviors of hypergeometric integrals associated with symmetric arrangements of three-dimensional spheres. We show, in an explicit way, h...

In this paper, we studied the Picard–Fuchs systems and equations which appear in the theory of Gauss–Manin systems and connections associated with deformations of isolated singularities. Among other things, we describe some interesting pr...

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...

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...

In this paper, a new class of kernels of integral transforms of the Laplace convolution type that we named symmetrical Sonin kernels is introduced and investigated. For a symmetrical Sonin kernel given in terms of elementary or special functions, its...

The Bateman functions and the allied Havelock functions were introduced as solutions of some problems in hydrodynamics about ninety years ago, but after a period of one or two decades they were practically neglected. In handbooks, the Bateman functio...

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...

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 unawa...

An integral representation is found for hypergeometric coherent states. It contains a generalized hypergeometric function. An asymptotic expansion of hypergeometric coherent states near $z=1$ is constructed. This expansion is used to find asymptotic ei...

The goal of this article is to establish several new formulas and new results related to the Marichev-Saigo-Maeda fractional integral and fractional derivative operators which are applied on the $(p,q)$-extended Bessel function. The results are express...

Motivated by statistical mechanics contexts, we study the properties of the q-Laplace transform, which is an extension of the well-known Laplace transform. In many circumstances, the kernel function to evaluate certain integral forms has been studied...