scholarly journals APPLICATION OF CERTAIN FRACTIONAL CALCULUS OPERATORS IN STATISTICAL DISTRIBUTIONS

1996 ◽  
Vol 27 (1) ◽  
pp. 27-36
Author(s):  
R. K. RAINA ◽  
MAMTA BOLIA
Keyword(s):  
Fractional Calculus ◽  
Special Function ◽  
Special Functions ◽  
Characteristic Functions ◽  
Statistical Distributions ◽  
Multivariate Statistical ◽  
Fractional Calculus Operators

This paper deals with the application of certain classes of fractional calculus operators in statistical distributions. The images of product combinations of special functions under the calculus of operators are applied to certain general­ ized forms of univariate and multivariate statistical distributions. Further results giving the expectations, cummulative functions and characteristic functions of such special function distributions are also obtained.

scholarly journals A Tutorial on the Basic Special Functions of Fractional Calculus

2020 ◽  
Vol 19 ◽  
Keyword(s):  
Probability Theory ◽  
Integral Equation ◽  
Fractional Calculus ◽  
Diffusion Equation ◽  
Special Function ◽  
Special Functions ◽  
Fractional Diffusion ◽  
Basic Properties ◽  
Fractional Relaxation ◽  
Time Fractional Diffusion Equation

In this tutorial survey we recall the basic properties of the special function of the Mittag-Leffler and Wright type that are known to be relevant in processes dealt with the fractional calculus. We outline the major applications of these functions. For the Mittag-Leffler functions we analyze the Abel integral equation of the second kind and the fractional relaxation and oscillation phenomena. For the Wright functions we distinguish them in two kinds. We mainly stress the relevance of the Wright functions of the second kind in probability theory with particular regard to the so-called M-Wright functions that generalizes the Gaussian and is related with the time-fractional diffusion equation.

scholarly journals The Marichev-Saigo-Maeda Fractional Calculus Operators Pertaining to the V -Function

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
S. Chandak ◽  
Biniyam Shimelis ◽  
Nigussie Abeye ◽  
A. Padma
Keyword(s):  
Fractional Calculus ◽  
Bessel Function ◽  
Hypergeometric Function ◽  
Exponential Function ◽  
Special Functions ◽  
Generalized Hypergeometric Function ◽  
Generalized Bessel Function ◽  
Fractional Calculus Operators

In the present paper, we establish some composition formulas for Marichev-Saigo-Maeda (MSM) fractional calculus operators with V -function as the kernel. In addition, on account of V -function, a variety of known results associated with special functions such as the Mittag-Leffler function, exponential function, Struve’s function, Lommel’s function, the Bessel function, Wright’s generalized Bessel function, and the generalized hypergeometric function have been discovered by defining suitable values for the parameters.

scholarly journals Fractional Integration of the Product of two Multivariable Gimel-Functions and a General Class of Polynomials

2018 ◽  
pp. 33-48
Author(s):  
Frederic Ayant
Keyword(s):  
Fractional Calculus ◽  
General Class ◽  
Special Functions ◽  
Fractional Integration ◽  
General Nature ◽  
Fractional Integral Operators ◽  
Summation Of Series ◽  
Fractional Calculus Operators ◽  
The Subject ◽  
Interesting Account

A significantly large number of earlier works on the subject of fractional calculus give the interesting account of the theory and applications of fractional calculus operators in many different areas of mathematical analysis (such as ordinary and partial differential equations, integral equations, special functions, the summation of series, etc.). The object of the present paper is to study and develop the Saigo-Maeda operators. First, we establish four results that give the images of the product of two multivariable Gimel-functions and a general class of multivariable polynomials in Saigo- Maeda operators. On account of the general nature of the Saigo-Maeda operators, multivariable Gimel-functions and a class multivariable polynomials a large number of new and known theorems involving Riemann-Liouville and Erdelyi- Kober fractional integral operators and several special functions.

scholarly journals New fractional approaches for n-polynomial P-convexity with applications in special function theory

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Shu-Bo Chen ◽  
Saima Rashid ◽  
Muhammad Aslam Noor ◽  
Zakia Hammouch ◽  
Yu-Ming Chu
Keyword(s):  
Fractional Calculus ◽  
Convex Functions ◽  
Special Functions ◽  
Real Life ◽  
Fractional Integrals ◽  
Digamma Function ◽  
Novel Approach ◽  
Inequality Theory ◽  
New Strategy ◽  
Significant Mechanism

Abstract Inequality theory provides a significant mechanism for managing symmetrical aspects in real-life circumstances. The renowned distinguishing feature of integral inequalities and fractional calculus has a solid possibility to regulate continuous issues with high proficiency. This manuscript contributes to a captivating association of fractional calculus, special functions and convex functions. The authors develop a novel approach for investigating a new class of convex functions which is known as an n-polynomial $\mathcal{P}$ P -convex function. Meanwhile, considering two identities via generalized fractional integrals, provide several generalizations of the Hermite–Hadamard and Ostrowski type inequalities by employing the better approaches of Hölder and power-mean inequalities. By this new strategy, using the concept of n-polynomial $\mathcal{P}$ P -convexity we can evaluate several other classes of n-polynomial harmonically convex, n-polynomial convex, classical harmonically convex and classical convex functions as particular cases. In order to investigate the efficiency and supremacy of the suggested scheme regarding the fractional calculus, special functions and n-polynomial $\mathcal{P}$ P -convexity, we present two applications for the modified Bessel function and $\mathfrak{q}$ q -digamma function. Finally, these outcomes can evaluate the possible symmetric roles of the criterion that express the real phenomena of the problem.

On some Hardy-type inequalities for fractional calculus operators

2017 ◽  
Vol 11 (2) ◽  
pp. 438-457 ◽  
Author(s):  
Sajid Iqbal ◽  
Josip Pečarić ◽  
Muhammad Samraiz ◽  
Zivorad Tomovski
Keyword(s):  
Fractional Calculus ◽  
Hardy Type Inequalities ◽  
Hardy Type ◽  
Fractional Calculus Operators
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