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This article is devoted to deriving a new linearization formula of a class for Jacobi polynomials that generalizes the third-kind Chebyshev polynomials class. In fact, this new linearization formula generalizes some existing ones in the literature. The derivation of this formula is based on employing a new moment formula of this class of polynomials and after that using suitable symbolic computation to reduce the resulting linearization coefficients into simplified forms that do not contain any hypergeometric functions or sums. The new formula is employed along with some other formulas and with the utilization of the spectral tau method to obtain numerical solutions to the nonlinear Fisher equation. The presented method is used to convert the equation governed by its underlying conditions into a nonlinear system of equations. The solution of the resulting system can be obtained through any suitable standard numerical scheme. To demonstrate the efficiency and usefulness of the proposed algorithm, some examples are shown, including comparisons with some existing techniques in the literature. [ABSTRACT FROM AUTHOR]

In this paper, for the proposed extended Laguerre polynomials A α q , n x , the generalized hypergeometric function of the type F q q , q > 2 and extension of the Laguerre polynomial are introduced. Similar to those related to the Laguerre polynomials, the generating function, recurrence relations, and Rodrigue's formula are determined. Some corollaries are also discussed at the end. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to establish several new closed-form evaluations of the generalized hypergeometric function F q + 1 q z for q = 2 , 3 , 4 , 5. This is achieved by means of separating the generalized hypergeometric function F q + 1 q z (q = 2 , 3 , 4 , 5) into even and odd components together with the use of several known infinite series involving reciprocals of the central binomial coefficients obtained earlier by Lehmer. [ABSTRACT FROM AUTHOR]

In 2003, Mercer presented an interesting variation of Jensen's inequality called Jensen–Mercer inequality for convex function. In the present paper, by employing harmonically convex function, we introduce analogous versions of Hermite–Hadamard inequalities of the Jensen–Mercer type via fractional integrals. As a result, we introduce several related fractional inequalities connected with the right and left differences of obtained new inequalities for differentiable harmonically convex mappings. As an application viewpoint, new estimates regarding hypergeometric functions and special means of real numbers are exemplified to determine the pertinence and validity of the suggested scheme. Our results presented here provide extensions of others given in the literature. The results proved in this paper may stimulate further research in this fascinating area. [ABSTRACT FROM AUTHOR]

In this paper, we compute the reproducing kernel B m , α z , w for the generalized Fock space F m , α 2 ℂ . The usual Fock space is the case when m = 2. We express the reproducing kernel in terms of a suitable hypergeometric series 1 F q . In particular, we show that there is a close connection between B 4 , α z , w and the error function. We also obtain the closed forms of B m , α z , w when m = 1 , 2 / 3 , 1 / 2. Finally, we also prove that B m , α z , z ~ e α z m z m − 2 as ∣ z ∣ ⟶ ∞. [ABSTRACT FROM AUTHOR]

Recently, hypergeometric functions of four variables are investigated by Bin-Saad and Younis. In this manuscript, our goal is to initiate a new quadruple hypergeometric function denoted by X 84 4 , and then, we ensure the existence of solutions of systems of partial differential equations for this function. We also establish some integral representations involving the quadruple hypergeometric function X 84 4 . [ABSTRACT FROM AUTHOR]

Generating functions plays an essential role in the investigation of several useful properties of the sequences which they generate. In this paper, we establish certain generating relations, involving some quadruple hypergeometric functions introduced by Bin-Saad and Younis. Some interesting special cases of our main results are also considered. [ABSTRACT FROM AUTHOR]

In this paper, by using certain inverse pairs of symbolic operators introduced by Choi and Hasanov in 2011, we establish several decomposition formulas associated with the Gaussian triple hypergeometric functions. Some transformation formulas for these functions have also been obtained. [ABSTRACT FROM AUTHOR]

In this work, we introduce the idea of n –polynomial harmonically s –type convex function. We elaborate the new introduced idea by examples and some interesting algebraic properties. As a result, new Hermite–Hadamard, some refinements of Hermite–Hadamard and Ostrowski type integral inequalities are established, which are the generalized variants of the previously known results for harmonically convex functions. Finally, we illustrate the applicability of this new investigation in special functions (hypergeometric function and special mean of real numbers). [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and uniqueness of positive solutions to a class of multipoint boundary value problems for singular fractional differential equations with the p -Laplacian operator. Here, the nonlinear source term f permits singularity with respect to its time variable t. Some fixed-point theorems such as the Leray-Schauder nonlinear alternative, the Schauder fixed-point theorem, and the Banach contraction mapping principle and the properties of the Gauss hypergeometric function are used to prove our main results. And by employing the upper and lower solutions technique, we derive a new approach to obtain the maximal and minimal solutions to the given problem. Finally, we present some examples to demonstrate our existence and uniqueness results. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new subclass of analytic functions in open unit disc. We obtain coefficient estimates, extreme points, and distortion theorem. We also derived the radii of close-to-convexity and starlikeness for this class. [ABSTRACT FROM AUTHOR]

We establish some new Hermite-Hadamard type integral inequalities for functions whose second-order mixed derivatives are coordinated (s,m)-P-convex. An expression form of Hermite-Hadamard type integral inequalities via the beta function and the hypergeometric function is also presented. Our results provide a significant complement to the work of Wu et al. involving the Hermite-Hadamard type inequalities for coordinated (s,m)-P-convex functions in an earlier article. [ABSTRACT FROM AUTHOR]

We introduce a new subclass of analytic functions in the unit disk U, defined by using the generalized hypergeometric functions, which extends some previous well-known classes defined by different authors. Inclusion results, radius problems, and some connections with the Bernardi-Libera-Livingston integral operator are discussed. [ABSTRACT FROM AUTHOR]