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The object of the present work is to deduce several important developments in various recursion relations, relevant differential recursion formulas, infinite summation formulas, integral representations, and integral operators for Horn’s hypergeometric functions Γ1 and Γ2.

The main object of this paper is to establish some fixed point results for F(ψ,ϕ)- contractions in partially-ordered metric spaces. As an application of one of these fixed point theorems, we discuss the existence of a unique solution for a coupled system of higher-order fractional differential equations with multi-point boundary conditions. The results presented in this paper are shown to extend many recent results appearing in the literature.

In this paper, we investigate the existence of a unique coupled fixed point for α−admissible mapping which is of F(ψ1, ψ2)−contraction in the context of M−metric space. We have also shown that the results presented in this paper would extend many recent results appearing in the literature. Furthermore, we apply our results to develop sufficient conditions for the existence and uniqueness of a solution for a coupled system of fractional hybrid differential equations with linear perturbations of second type and with three-point boundary conditions.

The main objective of this paper is to construct new analogous definitions of the families of Humbert functions using the generating function method as the starting point. We study a class of various results in the family of Humbert functions with the help of the families of generating functions, explicit representations, especially differential recurrence relations and study some of the significant properties of this family of functions.

The main aim of this work is to give a different approach to the proof of some properties for Ultraspherical matrix polynomials (UMPs). We obtain the connections between Laguerre, Hermite and Ultraspherical matrix polynomials. Some definitions of new families of Ultraspherical matrix polynomials are given. Finally, various families of linear, bilinear and bilateral generating matrix functions (GMFs) for UMPs are given.

The aim of this paper is to prove some common and coupled random fixed point theorems for a pair of weakly monotone random operators satisfying some rational type contraction in the setting of partially ordered S− metric space. Our results extend and generalize many existing results in the literature. Moreover, an example is given to support our results. Finally, the results are used to prove the existence and uniqueness of solution of some random functional equations.

In the present paper, we propose a multi-valued version of weakly mixed monotone property for two single-valued mappings in partially ordered S -metric spaces. Also, we state and prove some coupled fixed
point theorems using this property. These theorems extend the corresponding results in [10].

In this paper, we introduce the notion of mixed weakly monotone property for two hybrid pairs  and    each of them consists of multi-valued mapping    and single valued mapping     defined on partially ordered metric space and then we prove  coincidence and common fixed point theorems for two hybrid pairs under different contractive conditions. These theorems extend and generalize very recent results that can be found in [12] and many others.