On a family of the incomplete H-functions and associated integral transforms

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Manish Kumar Bansal ◽  
Devendra Kumar
Keyword(s):  
Integral Transform ◽  
Special Functions ◽  
Natural Extension ◽  
Integral Transforms ◽  
Legendre Transform ◽  
Physical Sciences ◽  
Special Cases ◽  
Improper Integrals ◽  
Jacobi Transform ◽  
Math Phys

Abstract Recently, Srivastava, Saxena and Parmar [H. M. Srivastava, R. K. Saxena and R. K. Parmar, Some families of the incomplete H-functions and the incomplete H ¯ {\overline{H}} -functions and associated integral transforms and operators of fractional calculus with applications, Russ. J. Math. Phys. 25 2018, 1, 116–138] suggested incomplete H-functions (IHF) that paved the way to a natural extension and decomposition of H-function and other connected functions as well as to some important closed-form portrayals of definite and improper integrals of different kinds of special functions of physical sciences. In this article, our key aim is to present some new integral transform (Jacobi transform, Gegenbauer transform, Legendre transform and 𝖯 δ {\mathsf{P}_{\delta}} -transform) of this family of incomplete H-functions. Further, we give several interesting new and known results which are special cases our key results.

scholarly journals Certain Integral Transform and Fractional Integral Formulas for the Generalized Gauss Hypergeometric Functions

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Junesang Choi ◽  
Praveen Agarwal
Keyword(s):  
Hypergeometric Functions ◽  
Fractional Integral ◽  
Integral Transform ◽  
Special Functions ◽  
Integral Transforms ◽  
Integral Formulas ◽  
Special Cases

A remarkably large number of integral transforms and fractional integral formulas involving various special functions have been investigated by many authors. Very recently, Agarwal gave some integral transforms and fractional integral formulas involving theFp(α,β)(·). In this sequel, using the same technique, we establish certain integral transforms and fractional integral formulas for the generalized Gauss hypergeometric functionsFp(α,β,m)(·). Some interesting special cases of our main results are also considered.

INTEGRAL TRANSFORM OF K4-FUNCTION

2021 ◽  
Vol 21 (2) ◽  
pp. 429-436
Author(s):  
SEEMA KABRA ◽  
HARISH NAGAR
Keyword(s):  
Laplace Transform ◽  
Hypergeometric Function ◽  
Integral Transform ◽  
Integral Transforms ◽  
Gauss Hypergeometric Function ◽  
Wright Function ◽  
Generalized Wright Function ◽  
Special Cases ◽  
Euler Transform

In this present work we derived integral transforms such as Euler transform, Laplace transform, and Whittaker transform of K4-function. The results are given in generalized Wright function. Some special cases of the main result are also presented here with new and interesting results. We further extended integral transforms derived here in terms of Gauss Hypergeometric function.

scholarly journals Geometric properties of some linear operators defined by convolution

2008 ◽  
Vol 39 (4) ◽  
pp. 325-334 ◽  
Author(s):  
R. Aghalary ◽  
A. Ebadian ◽  
S. Shams
Keyword(s):  
Analytic Functions ◽  
Unit Disc ◽  
Integral Transform ◽  
Integral Transforms ◽  
Linear Operators ◽  
Geometric Properties ◽  
Special Cases ◽  
Classical Integral ◽  
Alpha Beta

Let $\mathcal{A}$ denote the class of normalized analytic functions in the unit disc $ U $ and $ P_{\gamma} (\alpha, \beta) $ consists of $ f \in \mathcal{A} $ so that$ \exists ~\eta \in \mathbb{R}, \quad \Re \bigg \{e^{i\eta} \bigg [(1-\gamma) \Big (\frac{f(z)}{z}\Big )^{\alpha}+ \gamma \frac{zf'(z)}{f(z)} \Big (\frac{f(z)}{z}\Big )^{\alpha} - \beta\bigg ]\bigg \} > 0. $ In the present paper we shall investigate the integral transform$ V_{\lambda, \alpha}(f)(z) = \bigg \{\int_{0}^{1} \lambda(t) \Big (\frac{f(tz)}{t}\Big )^{\alpha}dt\bigg \}^{\frac{1}{\alpha}}, $ where $ \lambda $ is a non-negative real valued function normalized by $ \int_{0}^{1}\lambda(t) dt=1 $. Actually we aim to find conditions on the parameters $ \alpha, \beta, \gamma, \beta_{1}, \gamma_{1} $ such that $ V_{\lambda, \alpha}(f) $ maps $ P_{\gamma}(\alpha, \beta) $ into $ P_{\gamma_{1}}(\alpha, \beta_{1}) $. As special cases, we study various choices of $ \lambda(t) $, related to classical integral transforms.

scholarly journals On Generalized Fractional Integral Operators and the Generalized Gauss Hypergeometric Functions

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Dumitru Baleanu ◽  
Praveen Agarwal
Keyword(s):  
Hypergeometric Functions ◽  
Fractional Integral ◽  
Integral Transform ◽  
Special Functions ◽  
Fractional Integration ◽  
Fractional Integral Operators ◽  
Integral Formulas ◽  
Special Cases ◽  
Generalized Fractional Integral ◽  
Generalized Fractional Integral Operators

A remarkably large number of fractional integral formulas involving the number of special functions, have been investigated by many authors. Very recently, Agarwal (National Academy Science Letters) gave some integral transform and fractional integral formulas involving theFpα,β·. In this sequel, here, we aim to establish some image formulas by applying generalized operators of the fractional integration involving Appell’s functionF3(·)due to Marichev-Saigo-Maeda. Some interesting special cases of our main results are also considered.

From the hyper-Bessel operators of Dimovski to the generalized fractional calculus

2014 ◽  
Vol 17 (4) ◽  
Author(s):  
Virginia Kiryakova
Keyword(s):  
Fractional Calculus ◽  
Differential Operators ◽  
Integral Transform ◽  
Special Functions ◽  
Operational Calculus ◽  
Integral Transforms ◽  
Bessel Operators ◽  
Generalized Fractional Calculus ◽  
Basic Facts ◽  
Fox’S H Function

AbstractIn 1966 Ivan Dimovski introduced and started detailed studies on the Bessel type differential operators B of arbitrary (integer) order m ≥ 1. He also suggested a variant of the Obrechkoff integral transform (arising in a paper of 1958 by another Bulgarian mathematician Nikola Obrechkoff) as a Laplace-type transform basis of a corresponding operational calculus for B and for its linear right inverse integral operator L. Later, the developments on these linear singular differential operators appearing in many problems of mathematical physics, have been continued by the author of this survey who called them hyper-Bessel differential operators, in relation to the notion of hyper-Bessel functions of Delerue (1953), shown to form a fundamental system of solutions of the IVPs for By(t) = λy(t). We have been able to extend Dimovski’s results on the hyper-Bessel operators and on the Obrechkoff transform due to the happy hint to attract the tools of the special functions as Meijer’s G-function and Fox’s H-function to handle successfully these matters. These author’s studies have lead to the introduction and development of a theory of generalized fractional calculus (GFC) in her monograph (1994) and subsequent papers, and to various applications of this GFC in other topics of analysis, differential equations, special functions and integral transforms.Here we try briefly to expose the ideas leading to this GFC, its basic facts and some of the mentioned applications.

A family of the incomplete hypergeometric functions and associated integral transform and fractional derivative formulas

Filomat ◽  
2017 ◽  
Vol 31 (1) ◽  
pp. 125-140 ◽  
Author(s):  
Rekha Srivastava ◽  
Ritu Agarwal ◽  
Sonal Jain
Keyword(s):  
Hypergeometric Functions ◽  
Fractional Derivatives ◽  
Integral Transform ◽  
Special Functions ◽  
Applied Mathematics ◽  
Integral Transforms ◽  
Mathematical Physics ◽  
Natural Generalization ◽  
Derivative Formulas ◽  
Derivatives Of

Recently, Srivastava et al. [Integral Transforms Spec. Funct. 23 (2012), 659-683] introduced the incomplete Pochhammer symbols that led to a natural generalization and decomposition of a class of hypergeometric and other related functions as well as to certain potentially useful closed-form representations of definite and improper integrals of various special functions of applied mathematics and mathematical physics. In the present paper, our aim is to establish several formulas involving integral transforms and fractional derivatives of this family of incomplete hypergeometric functions. As corollaries and consequences, many interesting results are shown to follow from our main results.

The mellin integral transform in fractional calculus

2013 ◽  
Vol 16 (2) ◽  
Author(s):  
Yuri Luchko ◽  
Virginia Kiryakova
Keyword(s):  
Fractional Calculus ◽  
Fourier Transforms ◽  
Integral Transform ◽  
Special Functions ◽  
Integral Transforms ◽  
Fourier Integral ◽  
Monotone Functions ◽  
Completely Monotone ◽  
Mellin Convolution ◽  
New Applications

AbstractIn Fractional Calculus (FC), the Laplace and the Fourier integral transforms are traditionally employed for solving different problems. In this paper, we demonstrate the role of the Mellin integral transform in FC. We note that the Laplace integral transform, the sin- and cos-Fourier transforms, and the FC operators can all be represented as Mellin convolution type integral transforms. Moreover, the special functions of FC are all particular cases of the Fox H-function that is defined as an inverse Mellin transform of a quotient of some products of the Gamma functions.In this paper, several known and some new applications of the Mellin integral transform to different problems in FC are exemplarily presented. The Mellin integral transform is employed to derive the inversion formulas for the FC operators and to evaluate some FC related integrals and in particular, the Laplace transforms and the sin- and cos-Fourier transforms of some special functions of FC. We show how to use the Mellin integral transform to prove the Post-Widder formula (and to obtain its new modi-fication), to derive some new Leibniz type rules for the FC operators, and to get new completely monotone functions from the known ones.

On Self-reciprocal functions for Fourier-Bessel integral transforms

1961 ◽  
Vol 57 (4) ◽  
pp. 778-781
Author(s):  
Afzal Ahmad ◽  
V. Lakshmikanth
Keyword(s):  
Bessel Function ◽  
Integral Transform ◽  
Integral Transforms ◽  
Special Cases ◽  
Image Position

Following Hardy and Titchmarsh(1) a function f(x) is said to be self-reciprocal if it satisfies the Fourier-Bessel integral transformwhere Jp(x) is a Bessel function of order P ≥ –½. This integral is denoted by Rp. The special cases P ½ and P ½, we denote by Rs and Rc, respectively.

scholarly journals The H-function transform

1970 ◽  
Vol 11 (2) ◽  
pp. 142-148 ◽  
Author(s):  
K. C. Gupta ◽  
P. K. Mittal
Keyword(s):  
Laplace Transform ◽  
Inversion Formula ◽  
Integral Transform ◽  
Applied Mathematics ◽  
Integral Transforms ◽  
Hankel Transform ◽  
Stieltjes Transform ◽  
Special Cases ◽  
Mathematics And Physics

Here we introduce a new integral transform whose kernel is the H-function. Since most of the important functions occurring in Applied Mathematics and Physics are special cases of the H-function, various integral transforms involving these functions as kernels follow as special cases of our transform. We mention some of them here and observe that a study of this transform gives general and useful results which serve as key formulae for several important integral transforms viz. Laplace transform, Hankel transform. Stieltjes transform and the various generalizations of these transforms. In the end we establish an inversion formula for the new transform and point out its special cases which are generalizations of results found recently.

scholarly journals Computation of certain integral formulas involving generalized Wright function

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Nabiullah Khan ◽  
Talha Usman ◽  
Mohd Aman ◽  
Shrideh Al-Omari ◽  
Serkan Araci
Keyword(s):  
Hypergeometric Function ◽  
Special Functions ◽  
Gaussian Quadrature ◽  
Integral Transforms ◽  
Wright Function ◽  
Gaussian Quadrature Formula ◽  
Integral Formulas ◽  
Generalized Wright Function ◽  
Special Cases ◽  
Generalized Wright Hypergeometric Function

Abstract The aim of the paper is to derive certain formulas involving integral transforms and a family of generalized Wright functions, expressed in terms of the generalized Wright hypergeometric function and in terms of the generalized hypergeometric function as well. Based on the main results, some integral formulas involving different special functions connected with the generalized Wright function are explicitly established as special cases for different values of the parameters. Moreover, a Gaussian quadrature formula has been used to compute the integrals and compare with the main results by using graphical representations.

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