- Open Access
Some properties of Wright-type generalized hypergeometric function via fractional calculus
© Rao et al.; licensee Springer. 2014
- Received: 3 October 2013
- Accepted: 26 March 2014
- Published: 6 May 2014
This paper is devoted to the study of a Wright-type hypergeometric function (Virchenko, Kalla and Al-Zamel in Integral Transforms Spec. Funct. 12(1):89-100, 2001) by using a Riemann-Liouville type fractional integral, a differential operator and Lebesgue measurable real or complex-valued functions. The results obtained are useful in the theory of special functions where the Wright function occurs naturally.
MSC: 33C20, 33E20, 26A33, 26A99.
- fractional integral and differential operators
- generalized hypergeometric function
- Lebesgue measurable functions
where denominator parameters are neither zero nor negative integer.
If , then (3) reduces to a Gauss hypergeometric function .
Prajapati et al. , Prajapati and Shukla  and Srivastava et al.  used the fractional calculus approach in the study of an integral operator and also generalized the Mittag-Leffler function.
The subject of fractional calculus [17–20] deals with the investigations of integrals and derivatives of any arbitrary real or complex order, which unify and extend the notions of integer-order derivative and n-fold integral. It has gained importance and popularity during the last four decades or so, mainly due to its vast potential of demonstrated applications in various seemingly diversified fields of science and engineering, such as fluid flow, rheology, diffusion, relaxation, oscillation, anomalous diffusion, reaction-diffusion, turbulence, diffusive transport, electric networks, polymer physics, chemical physics, electro-chemistry of corrosion, relaxation processes in complex systems, propagation of seismic waves, dynamical processes in self-similar and porous structures. Recently some interesting results on fractional boundary value problems and fractional partial differential equations were also discussed by Nyamoradi et al.  and Baleanu et al. [22, 23].
where, ; , , ; .
First, we give preliminaries, notations and definitions.
Integration and differentiation of fractional order are traditionally defined by the left-sided Riemann-Liouville fractional integral operator and the right-sided Riemann-Liouville fractional integral operator and the corresponding Riemann-Liouville fractional derivative operators and [3, 17], which are given as follows.
is called the Riemann-Liouville left-sided fractional integral of order μ.
is called the Riemann-Liouville right-sided fractional integral of order μ.
respectively. Here denotes the maximal integer not exceeding real x.
This equation (15) easily reduces to the classical Riemann-Liouville fractional derivative operator when . Moreover, in its special case when , (15) reduces to the Caputo fractional derivative operator.
where, for ; for .
The following facts are prepared for our study.
Theorem 1.1 (Mathai and Haubold )
Theorem 1.2 (Srivastava and Manocha )
provided that , and .
where ; ; ; .
This completes the proof of (21).
This is the proof of (22).
This completes proof of the required assertion (23). □
is such that for chosen x and τ, .
- (iii)Riemann-Liouville fractional integrals of order μ for and can be easily got by using Theorem 1.2 as
, , , , .
This is a proof of the result. □
On putting , this reduces to .
this leads to the proof. □
This completes the proof of (28). □
holds for any summable function .
This is the proof of the first part of (30).
This is the proof of (31), and using the same procedure leads to the second identity of (30). □
We are very grateful to the anonymous referees for their careful reading and helpful comments.
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