Distributional Representation of a Special Fox–Wright Function with an Application
Abstract
:1. Introduction and Motivation
2. Preliminaries
2.1. Special Functions and Fractional Integral Transforms
2.2. Special Functions and Theory of Distributions
3. New Representation of a Fox–Wright Function with Application to the Fractional Kinetic Equation
3.1. New Fractional Image Formulae Involving a Fox–Wright Function
3.2. Generalized Fractional Derivatives Involving a Fox–Wright Function
4. Convergence of New Series Representation as a Distribution
4.1. Validity of the New Generalized Representation
4.2. New Properties of a Fox–Wright Function as a Distribution
- (a)
- The combined effect of a Fox–Wright function and any distribution is:
- (b)
- A Fox–Wright function multiplied by an arbitrary constantgives the following:
- (c)
- An arbitrary complex constantis used to shift a Fox–Wright function:
- (d)
- A Fox–Wright function is transposed as:
- (e)
- The independent variable multiplied by a positive constant :
- (f)
- Differentiating a Fox–Wright function as a distribution:
- (g)
- A special Fox–Wright function’s distributional Fourier transform:
- (h)
- The Fourier transform’s duality property:
- (i)
- The Fourier transform and Parseval’s identity:
- (j)
- Differentiation characteristics of the Fourier transform:
- (k)
- A Fox–Wright function’s Taylor series:
- (l)
- A Fox–Wright function has the property of convolution:
- (m)
- If is a bounded support distribution, then:
5. Further Applications and Discussion
6. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Cases of Equation (11) | Relationship among the Kernels of Different Fractional Operators [14] |
---|---|
Marichev–Saigo–Maeda (M-S-M) | |
Saigo | |
Erdélyi–Kober (E–K) | |
Riemann–Liouville (R–L) |
m = 3 | Marichev–Saigo–Maeda Fractional Integrals |
---|---|
m = 2 | Saigo fractional integrals |
m = 1 | Erdélyi–Kober fractional integrals |
m = 1 | Riemann–Liouville (R–L) fractional integrals |
m = 3 | Marichev–Saigo–Maeda Fractional Derivative |
---|---|
m = 2 | Saigo fractional derivative |
m = 1 | Erdélyi–Kober fractional derivative |
m = 1 | Riemann–Liouville (R–L) fractional derivative |
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Tassaddiq, A.; Srivastava, R.; Kasmani, R.M.; Almutairi, D.K. Distributional Representation of a Special Fox–Wright Function with an Application. Mathematics 2023, 11, 3372. https://doi.org/10.3390/math11153372
Tassaddiq A, Srivastava R, Kasmani RM, Almutairi DK. Distributional Representation of a Special Fox–Wright Function with an Application. Mathematics. 2023; 11(15):3372. https://doi.org/10.3390/math11153372
Chicago/Turabian StyleTassaddiq, Asifa, Rekha Srivastava, Ruhaila Md Kasmani, and Dalal Khalid Almutairi. 2023. "Distributional Representation of a Special Fox–Wright Function with an Application" Mathematics 11, no. 15: 3372. https://doi.org/10.3390/math11153372
APA StyleTassaddiq, A., Srivastava, R., Kasmani, R. M., & Almutairi, D. K. (2023). Distributional Representation of a Special Fox–Wright Function with an Application. Mathematics, 11(15), 3372. https://doi.org/10.3390/math11153372