Integrals Involving Aleph Function and Wright’s Generalized Hypergeometric Function

2017 ◽  
Vol 10 ◽  
pp. 20-26
Author(s):  
D.L. Suthar ◽  
Haile Habenom ◽  
Hagos Tadesse
Keyword(s):  
Hypergeometric Function ◽  
Generalized Hypergeometric Function ◽  
Special Cases ◽  
Special Case ◽  
Srivastava’S Polynomials

The aim of this paper is to establish certain integrals involving product of the Aleph function with Srivastava’s polynomials and Fox-Wright’s Generalized Hypergeometric function. Being unified and general in nature, these integrals yield a number of known and new results as special cases. For the sake of illustration, four corollaries are also recorded here as special case of our main results.

scholarly journals Further Extension of the Generalized Hurwitz-Lerch Zeta Function of Two Variables

Mathematics ◽  
2019 ◽  
Vol 7 (1) ◽  
pp. 48
Author(s):  
Kottakkaran Sooppy Nisar
Keyword(s):  
Zeta Function ◽  
Hypergeometric Function ◽  
Summation Formula ◽  
Integral Representations ◽  
Generalized Hypergeometric Function ◽  
Special Cases ◽  
Lerch Zeta Function ◽  
Function Of Two Variables

The main aim of this paper is to provide a new generalization of Hurwitz-Lerch Zeta function of two variables. We also investigate several interesting properties such as integral representations, summation formula, and a connection with the generalized hypergeometric function. To strengthen the main results we also consider some important special cases.

Extended Mittag-Leffler function and associated fractional calculus operators

2020 ◽  
Vol 27 (2) ◽  
pp. 199-209 ◽  
Author(s):  
Junesang Choi ◽  
Rakesh K. Parmar ◽  
Purnima Chopra
Keyword(s):  
Fractional Calculus ◽  
Hypergeometric Function ◽  
Hypergeometric Functions ◽  
Fractional Integrals ◽  
Generalized Hypergeometric Function ◽  
Pochhammer Symbol ◽  
Special Cases ◽  
Fractional Calculus Operators ◽  
Fractional Integrals And Derivatives ◽  
Appl Math

AbstractMotivated mainly by certain interesting recent extensions of the generalized hypergeometric function [H. M. Srivastava, A. Çetinkaya and I. Onur Kıymaz, A certain generalized Pochhammer symbol and its applications to hypergeometric functions, Appl. Math. Comput. 226 2014, 484–491] by means of the generalized Pochhammer symbol, we introduce here a new extension of the generalized Mittag-Leffler function. We then systematically investigate several properties of the extended Mittag-Leffler function including some basic properties, Mellin, Euler-Beta, Laplace and Whittaker transforms. Furthermore, certain properties of the Riemann–Liouville fractional integrals and derivatives associated with the extended Mittag-Leffler function are also investigated. Some interesting special cases of our main results are pointed out.

Integral transforms and probability distributions involving a generalized hypergeometric function

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Nabiullah Khan ◽  
Talha Usman ◽  
Mohd Aman ◽  
Shrideh Al-Omari ◽  
Junesang Choi
Keyword(s):  
Probability Distribution ◽  
Hypergeometric Function ◽  
Probability Distributions ◽  
Beta Function ◽  
Integral Transforms ◽  
Generalized Hypergeometric Function ◽  
Confluent Hypergeometric Functions ◽  
Special Cases ◽  
Legendre Transforms ◽  
Generalized Probability

Abstract Various extensions of the beta function together with their associated extended hypergeometric and confluent hypergeometric functions have been introduced and investigated. In this paper, using the very recently contrived extended beta function, we aim to introduce an extension F v p , q ; λ ; σ , τ u {{}_{u}F_{v}^{p,q;\lambda;\sigma,\tau}} of the generalized hypergeometric function F v u {{}_{u}F_{v}} and investigate certain classes of transforms and several identities of a generalized probability distribution involving this extension. In fact, we present some interesting formulas of Jacobi, Gegenbauer, pathway, Laplace, and Legendre transforms of this extension multiplied by a polynomial. We also introduce a generalized probability distribution to investigate its several related properties. Further, we consider some special cases of our main results with an argument about the derived process of a known result.

scholarly journals Unified Approach to Fractional Calculus Images of Special Functions—A Survey

Mathematics ◽  
2020 ◽  
Vol 8 (12) ◽  
pp. 2260 ◽  
Author(s):  
Virginia Kiryakova
Keyword(s):  
Fractional Calculus ◽  
Hypergeometric Function ◽  
Special Functions ◽  
Great Part ◽  
General Setting ◽  
Generalized Hypergeometric Function ◽  
Unified Approach ◽  
Generalized Fractional Calculus ◽  
Special Cases ◽  
Google Search

Evaluation of images of special functions under operators of fractional calculus has become a hot topic with hundreds of recently published papers. These are growing daily and we are able to comment here only on a few of them, including also some of the latest of 2019–2020, just for the purpose of illustrating our unified approach. Many authors are producing a flood of results for various operators of fractional order integration and differentiation and their generalizations of different special (and elementary) functions. This effect is natural because there are great varieties of special functions, respectively, of operators of (classical and generalized) fractional calculus, and thus, their combinations amount to a large number. As examples, we mentioned only two such operators from thousands of results found by a Google search. Most of the mentioned works use the same formal and standard procedures. Furthermore, in such results, often the originals and the images are special functions of different kinds, or the images are not recognized as known special functions, and thus are not easy to use. In this survey we present a unified approach to fulfill the mentioned task at once in a general setting and in a well visible form: for the operators of generalized fractional calculus (including also the classical operators of fractional calculus); and for all generalized hypergeometric functions such as pΨq and pFq, Fox H- and Meijer G-functions, thus incorporating wide classes of special functions. In this way, a great part of the results in the mentioned publications are well predicted and appear as very special cases of ours. The proposed general scheme is based on a few basic classical results (from the Bateman Project and works by Askey, Lavoie–Osler–Tremblay, etc.) combined with ideas and developments from more than 30 years of author’s research, and reflected in the cited recent works. The main idea is as follows: From one side, the operators considered by other authors are cases of generalized fractional calculus and so, are shown to be (m-times) compositions of weighted Riemann–Lioville, i.e., Erdélyi–Kober operators. On the other side, from each generalized hypergeometric function pΨq or pFq (p≤q or p=q+1) we can reach, from the final number of applications of such operators, one of the simplest cases where the classical results are known, for example: to 0Fq−p (hyper-Bessel functions, in particular trigonometric functions of order (q−p)), 0F0 (exponential function), or 1F0 (beta-distribution of form (1−z)αzβ). The final result, written explicitly, is that any GFC operator (of multiplicity m≥1) transforms a generalized hypergeometric function into the same kind of special function with indices p and q increased by m.

On Ostrowski type inequalities

2016 ◽  
Vol 56 (1) ◽  
pp. 5-27 ◽  
Author(s):  
Ravi P. Agarwal ◽  
Min-Jie Luo ◽  
R.K. Raina
Keyword(s):  
Hypergeometric Function ◽  
General Class ◽  
Fractional Integral ◽  
Integral Operators ◽  
Generalized Hypergeometric Function ◽  
Fractional Integral Operators ◽  
Special Cases

Abstract In this paper, new forms of Ostrowski type inequalities are established for a general class of fractional integral operators. The main results are used to derive Ostrowski type inequalities involving the familiar Riemann-Liouville fractional integral operators and other important integral operators. We further obtain similar types of inequalities for the integral operators whose kernels are the Fox-Wright generalized hypergeometric function. Several consequences and special cases of some of the results including applications to Stolarsky’s means are also pointed out.

scholarly journals Generalized Fractional Integral Operators Pertaining to the Product of Srivastava’s Polynomials and Generalized Mathieu Series

Mathematics ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 206 ◽  
Author(s):  
K.S. Nisar ◽  
D.L. Suthar ◽  
M. Bohra ◽  
S.D. Purohit
Keyword(s):  
Hypergeometric Function ◽  
Fractional Integral ◽  
Special Functions ◽  
Integral Operators ◽  
Factor X ◽  
Fractional Integral Operators ◽  
Special Cases ◽  
Generalized Fractional Integral ◽  
Mathieu Series ◽  
Srivastava’S Polynomials

Fractional calculus image formulas involving various special functions are important for evaluation of generalized integrals and to obtain the solution of differential and integral equations. In this paper, the Saigo’s fractional integral operators involving hypergeometric function in the kernel are applied to the product of Srivastava’s polynomials and the generalized Mathieu series, containing the factor x λ ( x k + c k ) − ρ in its argument. The results are expressed in terms of the generalized hypergeometric function and Hadamard product of the generalized Mathieu series. Corresponding special cases related to the Riemann–Liouville and Erdélyi–Kober fractional integral operators are also considered.

scholarly journals Unified integral associated with the generalized V-function

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
S. Chandak ◽  
S. K. Q. Al-Omari ◽  
D. L. Suthar
Keyword(s):  
Bessel Function ◽  
Hypergeometric Function ◽  
Exponential Function ◽  
Special Functions ◽  
General Nature ◽  
Generalized Hypergeometric Function ◽  
Generalized Bessel Function ◽  
Integral Formulas ◽  
Special Cases ◽  
Lommel Function

Abstract In this paper, we present two new unified integral formulas involving a generalized V-function. Some interesting special cases of the main results are also considered in the form of corollaries. Due to the general nature of the V-function, several results involving different special functions such as the exponential function, the Mittag-Leffler function, the Lommel function, the Struve function, the Wright generalized Bessel function, the Bessel function and the generalized hypergeometric function are obtained by specializing the parameters in the presented formulas. More results are also discussed in detail.

scholarly journals A new class of double integrals involving Generalized Hypergeometric Function 3F2

2020 ◽  
Vol 51 (1) ◽  
Author(s):  
Insuk Kim
Keyword(s):  
Hypergeometric Function ◽  
Research Paper ◽  
Generalized Hypergeometric Function ◽  
Double Integral ◽  
New Class ◽  
Special Cases ◽  
Summation Theorem ◽  
Double Integrals

The aim of this research paper is to evaluate fifty double integrals invoving generalized hypergeometric function (25 each) in the form of\begin{align*}\int_{0}^{1}\int_{0}^{1} & x^{c-1}y^{c+\al-1} (1-x)^{\al- 1}(1-y)^{\be-1}\, (1-xy)^{c+\ell-\al-\be+1}\;\\ &\times \;{}_3F_2 \left[\begin{array}{c}a,\,\,\,\,\,b,\,\,\,\,\,2c+\ell+ 1 \\ \frac{1}{2}(a+b+i+1),\,\,2c+j \end{array}; xy\right]\,dxdy\end{align*}and\begin{align*}\int_{0}^{1}\int_{0}^{1} & x^{c+\ell}y^{c+\ell+\al} (1-x)^{\al-1}(1-y)^{\be-1}\, (1-xy)^{c- \al-\be}\;\\ &\times \;{}_3F_2 \left[\begin{array}{c}a,\,\,\,\,\,b,\,\,\,\,\,2c+\ell+ 1 \\ \frac{1}{2}(a+b+i+1),\,\,2c+j \end{array}; 1-xy\right]\,dxdy\end{align*}in the most general form for any $\ell \in \mathbb{Z}$ and $i, j = 0, \pm 1, \pm2$.The results are derived with the help of generalization of Edwards's well known double integral due to Kim, {\it et al.} and generalized classical Watson's summation theorem obtained earlier by Lavoie, {\it et al}.More than one hundred ineteresting special cases have also been obtained.

scholarly journals A New Class of Integrals Involving Generalized Hypergeometric Function and Multivariable Aleph-Function

2020 ◽  
Vol 44 (4) ◽  
pp. 539-550
Author(s):  
DINESH KUMAR ◽  
FRÉDÉRIC AYANT ◽  
DEVENDRA KUMAR
Keyword(s):  
Hypergeometric Function ◽  
Summation Formula ◽  
Generalized Hypergeometric Function ◽  
New Class ◽  
Special Cases

The aim of this paper is to evaluate an interesting integral involving generalized hypergeometric function and the multivariable Aleph-function. The integral is evaluated with the help of an integral involving generalized hypergeometric function obtained recently by Kim et al. [?]. The integral is further used to evaluate an interesting summation formula concerning the multivariable Aleph-function. A few interesting special cases and corollaries have also been discussed.

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