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The extension of interval-valued and real-valued functions known as fuzzy interval-valued function (FIVF) has made substantial contributions to the theory of interval analysis. In this article, we explore the importance of h-Godunova-Levin fuzzy convex and preinvex functions and also develop the new generation of the Hermite-Hadamard and trapezoid-type fuzzy fractional integral by the implementation of generalized fuzzy fractional operators having modified version of the Bessel-Maitland E1vBMF function as its kernel. Moreover, we extract some well-known inequalities from our main results.

The main goal of this paper is to describe the new version of extended Bessel–Maitland function and discuss its special cases. Then, using the aforementioned function as their kernels, we develop the generalized fractional integral and differential operators. The convergence and boundedness of the newly operators and compare them with the existing operators such as the Saigo and Riemann–Liouville fractional operators are explored. The integral transforms of newly defined and generalized fractional operators in terms of the generalized Fox–Wright function are presented. Additionally, we discuss a few exceptional cases of the main result.

The main objective of this article is to establish generalized fractional Hermite–Hadamard and related type integral inequalities for h-Godunova–Levin convexity and h-Godunova–Levin preinvexity with extended Wright generalized Bessel function acting as kernel. Moreover, Hermite–Hadamard-type and trapezoid-type inequalities for several known convexities including Godunova–Levin function, classical convex, s-Godunova–Levin function, P-function, and s-convex function are deduced as corollaries. These obtained results are analyzed in the form of generalization of fractional inequalities.

In this paper, a new confluent hypergeometric gamma function and an associated confluent hypergeometric Pochhammer symbol are introduced. We discuss some properties, for instance, their different integral representations, derivative formulas, and generating function relations. Different special cases are also considered.

The aim of this paper is to prove some identities in the form of generalized Meijer G-function. We prove the relation of some known functions such as exponential functions, sine and cosine functions, product of exponential and trigonometric functions, product of exponential and hyperbolic functions, binomial expansion, logarithmic function, and sine integral, with the generalized Meijer G-function. We also prove the product of modified Bessel function of first and second kind in the form of generalized Meijer G-function and solve an integral involving the product of modified Bessel functions.

The major purpose of this paper is to use the fractional integral operator in terms of extended generalized Bessel function to estimate new fractional integral inequalities for the extended Chebyshev functional in the sense of synchronous functions. We prove a set of inequalities for the fractional integral operator in terms of extended generalized Bessel function integrals with one and two parameters. Also, we discussed some special cases of the obtained result.

Inspired by certain fascinating ongoing extensions of the special functions such as an extension of the Pochhammer symbol and generalized hypergeometric function, we present a new extension of the generalized Mittag-Leffler (ML) function εa,b;p,vκz1 in terms of the generalized Pochhammer symbol. We then deliberately find certain various properties and integral transformations of the said function εa,b;p,vκz1. Some particular cases and outcomes of the main results are also established.

In the article, we present several generalizations for the generalized Čebyšev type inequality in the frame of quantum fractional Hahn’s integral operator by using the quantum shift operator σΨqς=qς+1−qσς∈l1,l2,σ=l1+ω/1−q,0

In this present article, we establish certain new Pólya–Szegö-type tempered fractional integral inequalities by considering the generalized tempered fractional integral concerning another function Ψ in the kernel. We then prove certain new Chebyshev-type tempered fractional integral inequalities for the said operator with the help of newly established Pólya–Szegö-type tempered fractional integral inequalities. Also, some new particular cases in the sense of classical tempered fractional integrals are discussed. Additionally, examples of constructing bounded functions are considered. Furthermore, one can easily form new inequalities for Katugampola fractional integrals, generalized Riemann–Liouville fractional integral concerning another function Ψ in the kernel, and generalized fractional conformable integral by applying different conditions.

We solve the second-order linear differential equation called the k-hypergeometric differential equation by using Frobenius method around all its regular singularities. At each singularity, we find 8 solutions corresponding to the different cases for parameters and modified our solutions accordingly.

The main objective of the present paper is to define k-gamma and k-beta distributions and moments generating function for the said distributions in terms of a new parameter k>0. Also, the authors prove some properties of these newly defined distributions.