Fractional Calculus and Related InequalitiesView this Special Issue
Composition Formulae for the -Fractional Calculus Operators Associated with -Wright Function
In this article, the -fractional-order integral and derivative operators including the -hypergeometric function in the kernel are used for the -Wright function; the results are presented for the -Wright function. Also, some of special cases related to fractional calculus operators and -Wright function are considered.
1. Introduction and Preliminaries
Fractional calculus was introduced in 1695, but in the last two decades researchers have been able to use it properly on the account of availability of computational resources. In many areas of application of fractional calculus, the researchers found significant applications in science and engineering. In the literature, many applications of fractional calculus are available in astrophysics, biosignal processing, fluid dynamics, nonlinear control theory, stochastic dynamical system, and so on. Also, a number of researchers [1–10] have studied in-depth level of properties, applications, and various directions of extensions of Gauss hypergeometric function of fractional integration.
Recently, in a series of research publications on generalized classical fractional calculus operators, research by Mubeen and Habibullah  has been published on the integral part of the Riemann-Liouville version and its applications; an alternative definition for the -Riemann–Liouville fractional derivative was introduced by Dorrego . The left- and right-hand operators of Saigo -fractional integration and differentiation associated with the -Gauss hypergeometric function defined by Gupta and Parihar  (see also ) are as follows:where is the -Gauss hypergeometric function defined by  for :
The corresponding fractional differential operators have their respective forms aswhere , and is the integer part of .
Remark 1. If we set in equations (1), (2), (4), and (5), operators reduce to Saigo’s fractional integral and derivative operators stated in , respectively.
Now, we consider the following basic results for our study.
Lemma 1 (see , pp. 497, Eq. 4.2). Let ; then,
Lemma 2 (see , pp. 497, Eq. 4.3). Let , and ; then,
Lemma 3 (see , pp. 500, Eq. 6.2). Let such that ; then,
Lemma 4 (see , pp. 500, Eq. 6.3). Let and , and ; then,
Recently, Gehlot and Prajapati  studied the concept of the generalized -Wright function, which is presented in the following definition, and its connection with other special functions. It is the generalization of Mittag-Leffler function and many other special functions (see also, [16–21]). These special functions have found many important applications in solving problems of physics, biology, engineering, and applied sciences.
The -Wright function is defined for , , and aswith the convergence conditions described as
Remark 2. When we put in (10), the -Wright function reduces to Wright function which is stated in .
The following relation of the -Wright function in terms of the generalized -Mittag-Leffler function, -Bessel function, -hypergeometric function, and Mittag-Leffler family function is defined as follows by giving the appropriate values of the parameters:(1)For , the generalized -Mittag-Leffler function from Gehlot  is Here, Díaz and Pariguan  introduced the -Pochhammer symbol and -gamma function as follows: and the relation with classical Euler’s gamma function is as follows: where , and . For more information on the -Pochhammer symbol, -special functions, and fractional Fourier transforms, refer to Romero and Cerutti’s  articles.(2)For , the generalized -Mittag-Leffler function from Dorrego and Cerutti  is(3)For , the -Bessel function of the first kind from Cerutti  is(4)For , the -hypergeometric function with three parameters from Mubeen et al.  is(5)For , the generalized Mittag-Leffler function from Shukla and Prajapati  is(6)For , the generalized Mittag-Leffler function from Prabhakar  is(7)For , the Mittag-Leffler function from Wiman  is(8)For , the Mittag-Leffler function from Mittag-Leffler  is
2. Saigo -Fractional Integration in terms of -Wright Function
Theorem 1. Let , , and such that , . If condition (11) is satisfied and be the left-sided integral operator of the generalized -fractional integration associated with -Wright function, then the following equation holds true:
Proof. We indicate the R.H.S. of equation (22) by , and invoking equation (10), we obtainNow applying equation (6), we getNow, interpreting definition (10) on the aforementioned equation, we arrive at the desired result (22).
Theorem 2. Let , , and such that and , with . If condition (11) is satisfied and be the right-sided integral operator of the generalized -fractional integration associated with -Wright function, then the following equation holds true:
Proof. The finding is similar to that of Theorem 1. So, we omit the details.
3. Saigo -Fractional Differentiation in terms of -Wright Function
Theorem 3. Let , , and such that , . If condition (11) is satisfied and be the left-sided differential operator of the generalized -fractional differentiation associated with -Wright function, then the following equation holds true:
Proof. For simplicity, let denote the left side of (28). Using definition (10), we obtainNow, applying equation (8), we obtainIn accordance with (10), the required result is (28). This completes the proof of Theorem 3.
Theorem 4. Let , , and such that , , where . If condition (11) is satisfied and be the right-sided differential operator of the generalized -fractional differentiation associated with -Wright function, then thefollowing equation holds true:
Proof. The proof is parallel to that of Theorem 3. Therefore, we omit the details.
4. Special Cases and Concluding Remarks
Being very general, the results given in (22), (25), (28), and (31) can yield a wide number of special cases by assigning some appropriate values to the parameters involved. Now, as shown in the following, we are explaining a few corollaries.
The advantage of the generalized -fractional calculus operators, which are also called by many authors as the general operator, is that they generalize Saigo’s fractional calculus operators and classical Riemann–Liouville (R-L) operators. For , operators (1), (2), (4), and (5) reduce to Saigo’s  fractional integral and differentiation operators. If we take , (1), (2), (4), and (5) reduce the operators to -Riemann–Liouville as follows:
Due to the most general character of the -Wright function, numerous other interesting special cases from (22), (25), (28), and (31) can be given in the form of -Struve function, -Wright-type function, and many more, but due to lack of space, they are not represented here.
No data were used to support this study.
Conflicts of Interest
The author declares that there are no conflicts of interest.
G. A. Anastassiou, “Intelligent comparisons: analytic inequalities,” Studies in Computational Intelligence, vol. 609, Springer, Cham, Switzerland, 2016.View at: Google Scholar
L. Galué, S. L. Kalla, and T. V. Kim, “Composition of Erdélyi-Kober fractional operators,” Integral Transforms and Special Functions, vol. 9, no. 3, pp. 185–196, 2000.View at: Google Scholar
V. Kiryakova, “A brief story about the operators of the generalized fractional calculus,” Fractional Calculus and Applied Analysis, vol. 11, no. 2, pp. 203–220, 2008.View at: Google Scholar
M. Saigo, “A remark on integral operators involving the Gauss hypergeometric functions,” Mathematical Reports of College of General Education, vol. 11, pp. 135–143, 1978.View at: Google Scholar
M. Saigo, “A certain boundary value problem for the Euler-Darboux equation I,” Math Japonica, vol. 24, pp. 377–385, 1979.View at: Google Scholar
M. Saigo, “A certain boundary value problem for the Euler-Darboux equation II,” Math Japonica, vol. 24, pp. 211–220, 1980.View at: Google Scholar
R. K. Saxena, J. Ram, and D. L. Suthar, “Generalized fractional calculus of the generalized Mittag- Leffler functions,” The Journal of the Indian Academy of Mathematics, vol. 31, no. 1, pp. 165–172, 2009.View at: Google Scholar
S. Mubeen and G. M. Habibullah, “-fractional integrals and application,” International Journal of Contemporary Mathematical Sciences, vol. 7, no. 2, pp. 89–94, 2012.View at: Google Scholar
A. Gupta and C. L. Parihar, “Saigo’s -Fractional calculus operators,” Malaya Journal of Matematik, vol. 5, no. 3, pp. 494–504, 2017.View at: Google Scholar
D. L. Suthar, D. Baleanu, S. D. Purohit, and F. Uçar, “Certain -fractional calculus operators and image formulas of -Struve function,” AIMS Mathematics, vol. 5, no. 3, pp. 1706–1719, 2020.View at: Google Scholar
K. S. Gehlot and J. C. Prajapati, “Fractional calculus of generalized -Wright function,” Journal of Fractional Calculus and Applications, vol. 4, no. 2, pp. 283–289, 2013.View at: Google Scholar
G. A. Dorrego and R. A. Cerutti, “The -Mittag-Leffer function,” International Journal of Contemporary Mathematical Sciences, vol. 7, pp. 705–716, 2012.View at: Google Scholar
K. S. Gehlot, “The generalized -Mittag-Leffer function,” International Journal of Contemporary Mathematical Sciences, vol. 7, pp. 2213–2219, 2012.View at: Google Scholar
G. M. Mittag-Leffler, “Sur la nouvelle function .,” Comptes Rendus de l’Académie des Sciences, vol. 137, pp. 554–558, 1903.View at: Google Scholar
T. R. Prabhakar, “A singular integral equation with a generalized Mittag-Leffler function in the kernel,” Yokohama Mathematical Jounal, vol. 19, pp. 7–15, 1971.View at: Google Scholar
R. Díaz and E. Pariguan, “On hypergeometric functions and Pochhammer -symbol,” Divulgaciones Matemáticas, vol. 15, no. 2, pp. 179–192, 2007.View at: Google Scholar
L. Romero and R. Cerutti, “Fractional fourier transform and special -function,” International Journal of Contemporary Mathematical Sciences, vol. 7, no. 4, pp. 693–704, 2012.View at: Google Scholar
R. A. Cerutti, “On the -Bessel functions,” International Mathematical Forum, vol. 7, no. 38, pp. 1851–1857, 2012.View at: Google Scholar