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

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]

For a prime p ≡ 3 (mod 4) and a positive integer t, let q = p 2 t . Let g be a primitive element of the finite field F q . The Peisert graph P ∗ (q) is defined as the graph with vertex set F q where ab is an edge if and only if a - b ∈ ⟨ g 4 ⟩ ∪ g ⟨ g 4 ⟩ . We provide a formula, in terms of finite field hypergeometric functions, for the number of complete subgraphs of order four contained in P ∗ (q) . We also give a new proof for the number of complete subgraphs of order three contained in P ∗ (q) by evaluating certain character sums. The computations for the number of complete subgraphs of order four are quite tedious, so we further give an asymptotic result for the number of complete subgraphs of any order m in Peisert graphs. [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 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]

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. [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]

Hypergeometric functions of one and many variables play an important role in various branches of modern physics and mathematics. We often encounter hypergeometric functions with indices linearly dependent on a small parameter with respect to which we need to perform Laurent expansions. Moreover, it is desirable that such expansions be expressed in terms of well-known functions that can be evaluated with arbitrary precision. To solve this problem, we use the method of differential equations and the reduction of corresponding differential systems to a canonical basis. In this paper, we are interested in the generalized hypergeometric functions of one variable and in the Appell and Lauricella functions and their expansions in terms of the Goncharov polylogarithms. Particular attention is paid to the case of rational indices of the considered hypergeometric functions when the reduction to the canonical basis involves a nontrivial variable change. The paper comes with a Mathematica package Diogenes, which provides an algorithmic implementation of the required steps. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to establish as many as thirty new closed-form evaluations of the generalized hypergeometric function q+1Fq(z) for q = 2; 3. This is achieved by means of separating the generalized hypergeometric function q+1Fq(z) for q = 1; 2; 3 into even and odd components together with the use of several known infinite series involving reciprocal of the non-central binomial coefficients obtained earlier by L. Zhang and W. Ji. [ABSTRACT FROM AUTHOR]

We derive the recursion formulas for Horn hypergeometric functions H1 to H7. These recursion formulas help us write these functions as a combination of themselves. [ABSTRACT FROM AUTHOR]

The paper deals with the problem of representation of Horn's hypergeometric functions via continued fractions and branched continued fractions. We construct the formal continued fraction expansions for three ratios of Horn's hypergeometric functions H 7. The method employed is a two-dimensional generalization of the classical method of constructing a Gaussian continued fraction. It is proved that the continued fraction, which is an expansion of each ratio, uniformly converges to a holomorphic function of two variables on every compact subset of some domain of C 2 , and that this function is an analytic continuation of such a ratio in this domain. To illustrate this, we provide some numerical experiments at the end. [ABSTRACT FROM AUTHOR]

We present in this paper a generalization of the fractional kinetic equation using the generalized incomplete Wright hypergeometric function. The pathway-type transform technique is then used to investigate the solutions to a fractional kinetic equation with specific fractional transforms. Furthermore, exceptional cases of our outcomes are discussed and graphically illustrated using MATLAB software. This work provides a thorough overview for further investigation into these topics in order to gain a better understanding of their implications and applications. [ABSTRACT FROM AUTHOR]

In this paper, we derive some Fourier transforms of confluent hypergeometric functions. We give generalizations of several well-known results involving Fourier transforms of gamma functions. In particular, the generalizations include some Ramanujan's remarkable formulas. [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]

Kampé de Fériet hypergeometric functions are two-variable hypergeometric functions, which are a generalization of Appell's functions. It is known that they satisfy many reduction and summation formulas. In this paper, we define Kampé de Fériet hypergeometric functions over finite fields and show analogous formulas. [ABSTRACT FROM AUTHOR]

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 confluent hypergeometric function. Among the numerous extended hypergeometric functions, the primary rationale for choosing H 4 and X 8 is their comparable extension type. Next, we present various integral representations of Euler and Laplace types, Mellin and inverse Mellin transforms, Laguerre polynomial representations, transformation formulae, and a recurrence relation for the extended functions. In particular, we provide a generating function for X 8 , p and several bounding inequalities for H 4 , p and X 8 , p . We explore the utilization of the H 4 , p function within a probability distribution. Most special functions, such as the generalized hypergeometric function, the Beta function, and the p-extended Beta integral, exhibit natural symmetry. [ABSTRACT FROM AUTHOR]

The purpose of this article is to study how q-real numbers can be used for computations of convergence regions, q-integral representations of certain multiple triple q-Lauricella functions. The corresponding q-difference equations are also given without proof. In the process, we slightly improve Exton's original formulas. We also survey the current attempts to generalize the above functions to triple and quadruple hypergeometric functions. Finally, we compute some q-analogues of transformation formulas for Horn functions. [ABSTRACT FROM AUTHOR]

In this article, authors discussed about image formulas for the some new extended hypergeometric function using elementary properties of the fractional calculus integral operators. Furthermore, using integral transforms some image formulas are also established. [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]

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 conditions for the algebraic independence of such functions are established. [ABSTRACT FROM AUTHOR]

Recently Ono, Saad and the second author [21] initiated a study of value distribution of certain families of Gaussian hypergeometric functions over large finite fields. They investigated two families of Gaussian hypergeometric functions and showed that they satisfy semicircular and Batman distributions. Motivated by their results we aim to study distributions of certain families of hypergeometric functions in the p-adic setting over large finite fields. In particular, we consider two and six parameters families of hypergeometric functions in the p-adic setting and obtain that their limiting distributions are semicircular over large finite fields. In the process of doing this we also express the traces of pth Hecke operators acting on the spaces of cusp forms of even weight k ≥ 4 and levels 4 and 8 in terms of p-adic hypergeometric function which is of independent interest. These results can be viewed as p-adic analogous of some trace formulas of [1-2, 6]. [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]

Let A consists of analytic functions f:픻→ℂ satisfying f(0)=f′(0)−1=0. Let SNe∗ be the recently introduced Ma–Minda type functions family associated with the two‐cusped kidney‐shaped nephroid curve (u−1)2+v2−493−4v23=0 given by SNe∗:=f∈A:zf′(z)f(z)≺φNe(z)=1+z−z3/3. In this paper, we adopt a novel technique that uses the geometric properties of hypergeometric functions to determine sharp estimates on β so that each of the differential subordinations p(z)+βzp′(z)≺1+z;1+z;ez;imply p (z) ≺ φNe(z), where p (z) is analytic satisfying p(0)=1. As applications, we establish conditions that are sufficient to deduce that f∈A is a member of SNe∗. [ABSTRACT FROM AUTHOR]

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 functions, while the L-loop ladder integrals are related to the generalised hypergeometric L+1FL functions. [ABSTRACT FROM AUTHOR]

Many product formulas are known classically for generalized hypergeometric functions over the complex numbers. In this paper, we establish some analogous formulas for generalized hypergeometric functions over finite fields. [ABSTRACT FROM AUTHOR]

We prove two transformations for the p-adic hypergeometric function, which can be described as analogues of a transformation of Euler and a transformation of Clausen. We first evaluate certain character sums and then relate them to the p-adic hypergeometric functions to deduce the transformations. We use a character sum identity proved by Ahlgren, Ono, and Penniston to deduce the p-adic Clausen's transformation. We also deduce special values of certain p-adic hypergeometric functions. [ABSTRACT FROM AUTHOR]

The non-null distribution of the sample correlation coefficient under bivariate normality is derived when each of the associated two sample variances is subject to stripe truncation including usual single and double truncation as special cases. The probability density function is obtained using series expressions as in the untruncated case with new definitions of weighted hypergeometric functions. Formulas of the moments of arbitrary orders are given using the weighted hypergeometric functions. It is shown that the null joint distribution of the sample correlation coefficients under multivariate untruncated normality holds also in the variance-truncated cases. Some numerical illustrations are shown. [ABSTRACT FROM AUTHOR]

We sum in a closed form the Sneddon-Bessel series ∑∞ m=1 Jα(xjm,v)Jβ(yjm,v) /j m,v2n+α+β-2v+2 Jv+1(jm,v)², where 0 < x, 0 < y, x + y < 2, n is an integer, α, β, v ε C\{-1,-2, ... } with 2 Re v < 2n + 1 + Re α + Re β and {jm,v}m≥0 are the zeros of the Bessel function Jv of order v. In most cases, the explicit expressions for these sums involve hypergeometric functions pFq. As an application, we prove some extensions of the Kneser-Sommerfeld expansion. For instance, we show that ∑∞ m=1 j v-β m,v Jv (xjm,v)Jβ (yjm,v) (j²m,v-z²)Jv+1(jm,v)² = πJβ (yz)/4zβ-v Jv (z) (Yv (z)Jv (xz) - Jv (z)Yv (xz)), if Re v < Re β + 1 and 0 < y ≤ x, x + y < 2 (here, Yv denotes the Bessel function of the second kind), which becomes the Kneser-Sommerfeld expansion when β = v. [ABSTRACT FROM AUTHOR]

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 matrix function and state some integral equations and differential formulas about it. Next, we obtain some results about fractional calculus of these extended incomplete Wright hypergeometric matrix functions. Finally, we discuss an application of the extended incomplete Wright hypergeometric matrix function in the kinetic equations. [ABSTRACT FROM AUTHOR]

Given real parameters a , b , c and integer shifts n 1 , n 2 , m , we consider the ratio R (z) = 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 Im R (x ± i 0) with x > 1 in terms of real hypergeometric polynomial P, beta density and the absolute value of the Gauss hypergeometric function. This allows us to construct explicit integral representations for R when the asymptotic behaviour at unity is mild and the denominator does not vanish. The results are illustrated with a large number of examples. [ABSTRACT FROM AUTHOR]

General reduction of the elliptic hypergeometric equation to the level of complex hypergeometric functions is described. The derived equation is generalized to the Hamiltonian eigenvalue problem for new rational integrable N -body systems emerging from particular degenerations of the elliptic Ruijsenaars and van Diejen models. [ABSTRACT FROM AUTHOR]

A new subclass of bi-close-to-convex functions associated with the generalized hypergeometric functions defined in ∆ = { z ∈ C : | z | < 1 } is introduced. The estimates for the general Taylor–Maclaurin coefficients of the functions in the introduced subclass are obtained by making use of Faber polynomial expansions. In particular, several previous results are generalized. [ABSTRACT FROM AUTHOR]

Inspired by the work by Brychkov and Saad who gave infinite summation formulas for Appell functions (of two variables) with the help of summation theorems, we, using a new inverse pair of symbolic operators, establish infinite summation formulas for four Lauricella functions of three variables. As an application of the results obtained, we present integral representation formulas. [ABSTRACT FROM AUTHOR]

Owing to the remarkable success of the hypergeometric functions of one variable, the authors present a study of some families of hypergeometric functions of two or more variables. These functions include (for example) the Kampé de Fériet-type hypergeometric functions in two variables and Srivastava’s general hypergeometric function in three variables. The main aim of this paper is to provide several (presumably new) transformation and summation formulas for appropriately specified members of each of these families of hypergeometric functions in two and three variables. The methodology and techniques, which are used in this paper, are based upon the evaluation of some definite integrals involving logarithmic functions in terms of Riemann’s zeta function, Catalan’s constant, polylogarithm functions, and so on. [ABSTRACT FROM AUTHOR]

The power series expansions of the hypergeometric functions p + 1 F p (a , b 1 , ... , b p ; c 1 , ... , c p ; z) converge either inside the unit disk | z | < 1 or outside this disk | z | > 1 . Nørlund's expansion in powers of z / (z - 1) converges in the half-plane R (z) < 1 / 2 . For arbitrary z 0 ∈ C , Bühring's expansion in inverse powers of z - z 0 converges outside the disk | z - z 0 | = max { | z 0 | , | z 0 - 1 | } . None of them converge on the whole indented closed unit disk | z | ≤ 1 , z ≠ 1 . In this paper, we derive new expansions in terms of rational functions of z that converge in different regions, bounded or unbounded, of the complex plane that contain the indented closed unit disk. We give either explicit formulas for the coefficients of the expansions or recurrence relations. The key point of the analysis is the use of multi-point Taylor expansions in appropriate integral representations of p + 1 F p (a , b 1 , ... , b p ; c 1 , ... , c p ; z) . We show the accuracy of the approximations by means of several numerical experiments. [ABSTRACT FROM AUTHOR]

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 Meijer's G function. By the application of our method to known transformation formulas, we derive about forty hypergeometric identities, the majority of which are believed to be new. [ABSTRACT FROM AUTHOR]

In this article, we aim to investigate various formulae for the (p , k) -analogues of Gauss hypergeometric functions, including the integral transforms and the operators of fractional calculus. All the outcomes presented here are of general attractiveness and can yield a number of previous works as special cases. [ABSTRACT FROM AUTHOR]

New asymptotic expansions are derived of the Kummer functions M (a , b , z) and U (a , b + 1 , z) for large positive values of a and b, with z fixed. For both functions we consider b / a ≤ 1 and b / a ≥ 1 , with special attention for the case a ∼ b. We use a uniform method to handle all cases of these parameters. [ABSTRACT FROM AUTHOR]

In this paper we establish some Turan type inequalities for the τ-hypergeometric functions. We further prove the monotonicity of ratios of the τ-hypergeometric functions and it is depicted how our results are closely connected with certain Turan type inequalities. Additionally, we prove some results which correspond to the sub-additivity properties of the t-Gauss hypergeometric functions. We also present an open problem which may be of interest for further investigation. [ABSTRACT FROM AUTHOR]

This article suggests a new hypergeometric function for function fields and a new operator such that the Drinfeld logarithm is stable under it, and studies an equation satisfied by the hypergeometric function. As applications, our function is related to a supersingular polynomial of Drinfeld modules, a period of Drinfeld modules, and the Kochubei's polylogarithm, which are function-field analogues of well-known facts for the classical setting. Moreover, we generalize results for the Carlitz modules proven by Thakur to those for the Drinfeld ones. [ABSTRACT FROM AUTHOR]

In this paper, certain subclass of harmonic multivalent function defined in the exterior of the unit disk by used generalize hypergeometric functions is introduced . In This study an attempting have been made to investigate several geometric properties such as coefficient property, growth bounds, extreme points, convolution property, and convex linear combination . [ABSTRACT FROM AUTHOR]

Motivated by the substantial development of the special functions, we contribute to establish some rigorous results on the general series identities with bounded sequences and hypergeometric functions with different arguments, which are generally applicable in nature. For the application purpose, we apply our results to some functions e.g. Trigonometric functions, Elliptic integrals, Dilogarithmic function, Error function, Incomplete gamma function, and many other special functions. [ABSTRACT FROM AUTHOR]

For a general number 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. [ABSTRACT FROM AUTHOR]

The main object of this paper is to establish seven elliptical integrals associated with hypergeometric functions and suggest new way to compute their numerical values. The results presented in this article are presumably new and not present in the scientific literature. [ABSTRACT FROM AUTHOR]

ferential equations in physics, chemistry, and engineering. Its two power series solutions are the Kummer function, M(a,b,z) often referred to as the confluent hypergeometric function of the first kind, and where a and b are parameters that appear in the differential equation. A third function, the Tricomi function, M(a, b, z), sometimes referred to as the confluent hypergeometric function of the second kind, is also a solution of the confluent hypergeometric equation that is routinely used. Contrary to common procedure, all three of these functions (and more) must be considered in a search for the two linearly independentsolutionsoftheconfluenthypergeometricequation. Therearesituations,when a and a-b are integers,where one of these functions is not defined, or two of the functions are not linearly independent, or one of the linearly independents ol utions of the differentia lequationis different from these three functions. Many of these special cases correspond precisely to cases needed to solve problems in physics. This leads to significant confusion about how to work with confluent hyper geometric equations, inspite of authoritative references such as the NIST Digital Library of Mathematical Functions. Here, we carefully describe all of the different cases one has to consider and what the explicit formulas are for the two linearly independent solutions of the confluent hypergeometric equation. The procedure to properly solve the confluent hypergeometric equation is summarized in a convenient table. As an example, we use these solutions to study the bound states of the hydrogenic atom, correcting the standard treatment in textbooks. We also briefly consider the cutoff Coulomb potential. We hope that this guide will aid physicists to properly solve problems that involve the confluent hypergeometric differential equation. [ABSTRACT FROM AUTHOR]

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, extensions of one and several variables, continued fractions, Riemann's equation, group of the hypergeometric equation, summation, and transformation formulae. Among the various approaches to these functions, the transformation formulae for the hypergeometric series 2 F 1 and the generalized hypergeometric series p F q are significant, both in terms of applications and theory. The purpose of this paper is to establish a number of transformation formulae for p F q , whose particular cases would include Gauss's and Kummer's quadratic transformation formulae for 2 F 1 , as well as their two extensions for 3 F 2 , by making advantageous use of a recently introduced sequence and some techniques commonly used in dealing with p F q theory. The p F q function, which is the most significant function investigated in this study, exhibits natural symmetry. [ABSTRACT FROM AUTHOR]

A remarkably large number of hypergeometric (and generalized) functions and a variety of their extensions have been presented and investigated in the literature by many authors. In this paper, we introduce five new hypergeometric functions in four variables and then establish several recursion formulas for these new functions. Some interesting particular cases and consequences of the main results are also considered. [ABSTRACT FROM AUTHOR]