On Modifications of the Gamma Function by Using Mittag-Leffler Function

Tassaddiq, Asifa; Alruban, Abdulrahman

doi:10.1155/2021/9991762

Cited by 5 publications

(3 citation statements)

References 39 publications

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“…We conclude that this work is more fruitful over the others presented in [32] and the references provided therein, because those contain only the delta function, but here, we use the fractional derivatives of the Dirac delta function. This study sheds more light on the potential of the new representations developed in the references [32][33][34] and their related works, to achieve a broader applicability compared to other results, for example, refer [35][36][37].…”

Section: Discussionmentioning

confidence: 76%

Complex Generalized Representation of Gamma Function Leading to the Distributional Solution of a Singular Fractional Integral Equation

Tassaddiq,

Srivastava,

Kasmani

et al. 2023

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Firstly, a basic question to find the Laplace transform using the classical representation of gamma function makes no sense because the singularity at the origin nurtures so rapidly that Γze−sz cannot be integrated over positive real numbers. Secondly, Dirac delta function is a linear functional under which every function f is mapped to f(0). This article combines both functions to solve the problems that have remained unsolved for many years. For instance, it has been demonstrated that the power law feature is ubiquitous in theory but challenging to observe in practice. Since the fractional derivatives of the delta function are proportional to the power law, we express the gamma function as a complex series of fractional derivatives of the delta function. Therefore, a unified approach is used to obtain a large class of ordinary, fractional derivatives and integral transforms. All kinds of q-derivatives of these transforms are also computed. The most general form of the fractional kinetic integrodifferential equation available in the literature is solved using this particular representation. We extend the models that were valid only for a class of locally integrable functions to a class of singular (generalized) functions. Furthermore, we solve a singular fractional integral equation whose coefficients have infinite number of singularities, being the poles of gamma function. It is interesting to note that new solutions were obtained using generalized functions with complex coefficients.

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Section: Discussionmentioning

confidence: 76%

Complex Generalized Representation of Gamma Function Leading to the Distributional Solution of a Singular Fractional Integral Equation

Tassaddiq,

Srivastava,

Kasmani

et al. 2023

Axioms

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“…Research involving Gamma function is conducted by many authors currently. Results concerning extensions of Gamma function involving Mittag-Leffler function are presented in [20]. Other extensions of Gamma function are investigated in [21] using the two-parameter Mittag-Leffler matrix function and some important properties of these extended matrix functions are proved.…”

Section: Introductionmentioning

confidence: 99%

An Asymptotic Expansion for the Generalized Gamma Function

2022

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The symmetric patterns that inequalities contain are reflected in researchers’ studies in many mathematical sciences. In this paper, we prove an asymptotic expansion for the generalized gamma function Γμ(v) and study the completely monotonic (CM) property of a function involving Γμ(v) and the generalized digamma function ψμ(v). As a consequence, we establish some bounds for Γμ(v), ψμ(v) and polygamma functions ψμ(r)(v), r≥1.

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“…A special class of symmetric operators is de ned by using some special functions, which are satisfying the symmetric behavior. e Mittag-Le er function and its extensions, including Raina's functions, are solutions for all categories of fractional di erential equations (see [2][3][4][5][6][7][8]).…”

Section: Introductionmentioning

confidence: 99%

A Class of Symmetric Fractional Differential Operator Formed by Special Functions

Aldawish

Ibrahim

Obaiys

2022

Journal of Mathematics

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In light of a certain sort of fractional calculus, a generalized symmetric fractional differential operator based on Raina’s function is built. The generalized operator is then used to create a formula for analytic functions of type normalized. We use the ideas of subordination and superordination to show a collection of inequalities using the suggested differential operator. The new Raina’s operator is also used to the generalized kinematic solutions (GKS). Using the concepts of subordination and superordination, we provide analytic solutions for GKS. As a consequence, a certain hypergeometric function provides the answer. A fractional coefficient differential operator is also created. The geometric and analytic properties of the object are being addressed. The symmetric differential operator in a complex domain is shown to be a generalized fractional differential operator. Finally, we explore the characteristics of the Raina’s symmetric differential operator.

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On Modifications of the Gamma Function by Using Mittag-Leffler Function

Cited by 5 publications

References 39 publications

Complex Generalized Representation of Gamma Function Leading to the Distributional Solution of a Singular Fractional Integral Equation

Complex Generalized Representation of Gamma Function Leading to the Distributional Solution of a Singular Fractional Integral Equation

An Asymptotic Expansion for the Generalized Gamma Function

A Class of Symmetric Fractional Differential Operator Formed by Special Functions

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