scholarly journals Some New Extensions on Fractional Differential and Integral Properties for Mittag-Leffler Confluent Hypergeometric Function

2021 ◽  
Vol 5 (4) ◽  
pp. 143
Author(s):  
F. Ghanim ◽  
Hiba F. Al-Janaby ◽  
Omar Bazighifan
Keyword(s):  
Fractional Calculus ◽  
Hypergeometric Function ◽  
Hypergeometric Functions ◽  
Fractional Integration ◽  
Confluent Hypergeometric Function ◽  
Experimental Results ◽  
Integral Expression ◽  
Confluent Hypergeometric Functions ◽  
Analytical Properties ◽  
Fractional Differential

This article uses fractional calculus to create novel links between the well-known Mittag-Leffler functions of one, two, three, and four parameters. Hence, this paper studies several new analytical properties using fractional integration and differentiation for the Mittag-Leffler function formulated by confluent hypergeometric functions. We construct a four-parameter integral expression in terms of one-parameter. The paper explains the significance and applications of each of the four Mittag-Leffler functions, with the goal of using our findings to make analyzing specific kinds of experimental results considerably simpler.

scholarly journals INTEGRALS OF CONFLUENT HYPERGEOMETRIC FUNCTIONS

1990 ◽  
Vol 21 (2) ◽  
pp. 131-135
Author(s):  
GIOVANNA PITTALUGA
Keyword(s):  
Hypergeometric Function ◽  
Hypergeometric Functions ◽  
Confluent Hypergeometric Function ◽  
Weight Functions ◽  
Confluent Hypergeometric Functions

The moments of the weight functions $w(x)=e^{-x}x^\mu(\ln x)^\rho$, $\rho=0, 1, 2$, on $[0, \infty)$ with respect to the Confluent Hypergeometric function $\phi(a - n, c;x)$, $n = 0, 1, 2, \cdots$, are explicitly evaluated.

scholarly journals On the integral representations for the confluent hypergeometric function

2016 ◽  
Vol 472 (2193) ◽  
pp. 20160421 ◽  
Author(s):  
Bujar Xh. Fejzullahu
Keyword(s):  
Integral Representation ◽  
Hypergeometric Function ◽  
Hypergeometric Functions ◽  
Confluent Hypergeometric Function ◽  
Integral Representations ◽  
Contour Integral ◽  
Confluent Hypergeometric Functions ◽  
Special Cases

In this paper, we derive a new contour integral representation for the confluent hypergeometric function as well as for its various special cases. Consequently, we derive expansions of the confluent hypergeometric function in terms of functions of the same kind. Furthermore, we obtain a new identity involving integrals and sums of confluent hypergeometric functions. Our results generalized several well-known results in the literature.

scholarly journals A study of generalized summation theorems for the series 2F1 with an applications to Laplace transforms of convolution type integrals involving Kummer's functions 1F1

2018 ◽  
Vol 12 (1) ◽  
pp. 257-272
Author(s):  
Gradimir Milovanovic ◽  
Rakesh Parmar ◽  
Arjun Rathie
Keyword(s):  
Hypergeometric Function ◽  
Hypergeometric Functions ◽  
Integral Transform ◽  
Laplace Transforms ◽  
Confluent Hypergeometric Function ◽  
Convolution Type ◽  
Product Theorem ◽  
Confluent Hypergeometric Functions

Motivated by recent generalizations of classical theorems for the series 2F1 [Integral Transform. Spec. Funct. 229(11), (2011), 823-840] and interesting Laplace transforms of Kummer's confluent hypergeometric functions obtained by Kim et al. [Math. Comput. Modelling 55 (2012), 1068-1071], first we express generalized summations theorems in explicit forms and then by employing these, we derive various new and useful Laplace transforms of convolution type integrals by using product theorem of the Laplace transforms for a pair of Kummer's confluent hypergeometric function.

scholarly journals Fractional-Order Derivatives Defined by Continuous Kernels: Are They Really Too Restrictive?

2020 ◽  
Vol 4 (3) ◽  
pp. 40
Author(s):  
Jocelyn Sabatier
Keyword(s):  
Fractional Calculus ◽  
Fractional Order ◽  
Fractional Differential Equations ◽  
Initial Conditions ◽  
Fractional Integration ◽  
Current Trend ◽  
Singular Kernels ◽  
Fractional Differential ◽  
Definition Of ◽  
A Current

In the field of fractional calculus and applications, a current trend is to propose non-singular kernels for the definition of new fractional integration and differentiation operators. It was recently claimed that fractional-order derivatives defined by continuous (in the sense of non-singular) kernels are too restrictive. This note shows that this conclusion is wrong as it arises from considering the initial conditions incorrectly in (partial or not) fractional differential equations.

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.

Application of the E-operator to evaluate an infinite integral

1969 ◽  
Vol 65 (3) ◽  
pp. 725-730 ◽  
Author(s):  
F. Singh
Keyword(s):  
Finite Difference ◽  
Hypergeometric Function ◽  
Hypergeometric Functions ◽  
General Nature ◽  
Difference Operators ◽  
Generalized Hypergeometric Function ◽  
Generalized Hypergeometric Functions ◽  
Confluent Hypergeometric Functions ◽  
Infinite Integral

1. The object of this paper is to evaluate an infinite integral, involving the product of H-functions, generalized hypergeometric functions and confluent hypergeometric functions by means of finite difference operators E. As the generalized hypergeometric function and H-function are of a very general nature, the integral, on specializing the parameters, leads to a generalization of many results some of which are known and others are believed to be new.

scholarly journals Two integral transform pairs involving hypergeometric functions

1965 ◽  
Vol 7 (1) ◽  
pp. 42-44 ◽  
Author(s):  
Jet Wimp
Keyword(s):  
Hypergeometric Function ◽  
Hypergeometric Functions ◽  
Integral Transform ◽  
Confluent Hypergeometric Function ◽  
Gaussian Hypergeometric Function ◽  
Special Cases

In this note, we first establish an integral transform pair where the kernel of each integral involves the Gaussian hypergeometric function. Special cases of Theorem 1 have been studied by several authors [1, 2, 5, 6]. In Theorem 2 a similar integral transform pair involving a confluent hypergeometric function is given.

scholarly journals Some Inequalities of Extended Hypergeometric Functions

Mathematics ◽  
2021 ◽  
Vol 9 (21) ◽  
pp. 2702
Author(s):  
Shilpi Jain ◽  
Rahul Goyal ◽  
Praveen Agarwal ◽  
Juan L. G. Guirao
Keyword(s):  
Hypergeometric Function ◽  
Integral Inequality ◽  
Hypergeometric Functions ◽  
Beta Function ◽  
Confluent Hypergeometric Function ◽  
Gauss Hypergeometric Function ◽  
Extended Type ◽  
Mathematical Sciences ◽  
Set Up ◽  
Extended Beta Function

Hypergeometric functions and their inequalities have found frequent applications in various fields of mathematical sciences. Motivated by the above, we set up certain inequalities including extended type Gauss hypergeometric function and confluent hypergeometric function, respectively, by virtue of Hölder integral inequality and Chebyshev’s integral inequality. We also studied the monotonicity, log-concavity, and log-convexity of extended hypergeometric functions, which are derived by using the inequalities on an extended beta function.

scholarly journals Transformation formulas of incomplete hypergeometric functions via fractional calculus operators

2021 ◽  
Vol 13(62) (2) ◽  
pp. 571-580
Author(s):  
Kamlesh Jangid ◽  
Sunil Dutt Purohit ◽  
Daya Lal Suthar
Keyword(s):  
Fractional Calculus ◽  
Hypergeometric Function ◽  
Present Article ◽  
Hypergeometric Functions ◽  
Fractional Calculus Operators ◽  
Transformation Formulas

The desire for present article is to derive from the application of fractional calculus operators a transformation that expresses a potentially useful incomplete hypergeometric function in various forms of a countable sum of lesser-order functions. Often listed are numerous (known or new) specific cases and implications of the findings described herein

scholarly journals Certain investigations in the field of special functions

2020 ◽  
Vol 1 (1) ◽  
pp. 87-98
Author(s):  
Maisoon A. Kulib ◽  
Ahmed A. Al-Gonah ◽  
Salem S. Barahmah
Keyword(s):  
Hypergeometric Function ◽  
Special Functions ◽  
Beta Function ◽  
Confluent Hypergeometric Function ◽  
Integral Representations ◽  
Confluent Hypergeometric Functions ◽  
Research Fields ◽  
Summation Formulas ◽  
Physical Problems ◽  
Wide Range

Motivated mainly by a variety of applications of Euler's Beta, hypergeometric, and confluent hypergeometric functions together with their extensions in a wide range of research fields such asengineering, chemical, and physical problems. In this paper, we introduce modified forms of some extended special functions such as Gamma function, Beta function, hypergeometric function and confluent hypergeometric function by making use of the idea given in reference \cite{9}. Also, certain investigations including summation formulas, integral representations and Mellin transform of these modified functions are derived. Further, many known results are obtained asspecial cases of our main results.

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