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Some properties of Wright-type generalized hypergeometric function via fractional calculus
Advances in Difference Equations volume 2014, Article number: 119 (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.
1 Introduction and preliminaries
The Gauss hypergeometric function is defined  as
The generalized hypergeometric function in a classical sense has been defined  as
where denominator parameters are neither zero nor negative integer.
Several generalizations of hypergeometric functions [6–13]etc. have been made and also motivated us to further investigate the topic. Virchenko et al.  defined the generalized hypergeometric function in a different manner (throughout the paper, we call this function the Wright-type generalized hypergeometric function) as follows:
If , then (3) reduces to a Gauss hypergeometric function .
Rao et al.  obtained many properties for the function as defined in (3) including the following result. If ; , , , then
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].
In continuation of the study on the significance of fractional calculus, we define the integral operator as follows:
where, ; , , ; .
Substituting , (5) reduces to the operator
First, we give preliminaries, notations and definitions.
is the space of Lebesgue measurable real or complex-valued functions such that
The Gauss multiplication formula  is given as follows. If m is a positive integer and , then
The representation of a generalized factorial function in terms of the Pochhammer symbol  is given for
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.
If , , , then
is called the Riemann-Liouville left-sided fractional integral of order μ.
is called the Riemann-Liouville right-sided fractional integral of order μ.
For , ; , the left-sided and right-sided Riemann-Liouville fractional derivatives are defined as
respectively. Here denotes the maximal integer not exceeding real x.
A generalization of the Riemann-Liouville fractional derivative operator (13) has been made by introducing the fractional derivative operator of order and type with respect to x as follows :
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.
The left- and right-sided Caputo fractional derivatives of order (), denoted by and respectively, are defined on via the Riemann-Liouville fractional derivatives as
where, for ; for .
The following facts are prepared for our study.
Theorem 1.1 (Mathai and Haubold )
If , , , then
Theorem 1.2 (Srivastava and Manocha )
If a function , analytic in the disc , has the power series expansion (), then
provided that , and .
Lemma 1.1 The following result (Srivastava and Tomovski ) holds true for the fractional derivative operator defined by (13) as
where ; ; ; .
2 Main results
Theorem 2.1 If , , , , , , , then for , and τ, ,
If , , then
The use of (18) gives
This completes the proof of (21).
From (22) and (13), we get
and, using (21), this takes the following form:
Applying (4) gives
This is the proof of (22).
and using the identity (20) yields
This completes proof of the required assertion (23). □
Corollary If and , then
is such that for chosen x and τ, .
Proof The result can be obtained directly by multiplying (21) by λ and taking , , , . Remarks:
Riemann-Liouville fractional integrals of order μ for and can be easily got by using Theorem 1.2 as
Theorem 2.2 For , the generalized hypergeometric function takes the form
, , , , .
Proof Putting in (8), we obtain
From (26) and (3) afterwards, , we get
This is a proof of the result. □
On putting , this reduces to .
3 Some properties of the operator
Theorem 3.1 If ; , , ; ; , then
Proof From (5)
this leads to the proof. □
Theorem 3.2 If ; , , ; and , then the operator is bounded on and
Proof From (5) and (7), afterwards interchanging the order of integration by applying the Dirichlet formula , we obtain
and substituting , we have
Using (3) and further simplification gives
This equation can also be written as
This completes the proof of (28). □
Theorem 3.3 If ; , , ; and , then
holds for any summable function .
Proof From (11) and (5), we have
Interchanging the order of integration and using the Dirichlet formula , we get
Substituting , we get
Making the use of (11) and applying (21) yield
This is the proof of the first part of (30).
For proving the second part of the theorem, we start from the right-hand side of (30) and, using (5), we get
Using the Dirichlet formula  and interchanging the order of integration, we have
Substituting in the above equation, we get
This is the proof of (31), and using the same procedure leads to the second identity of (30). □
Kiryakova V: Generalized Fractional Calculus and Applications. Wiley, New York; 1994.
Mathai AM, Saxena RK: Generalized Hypergeometric Functions with Applications in Statistics and Physical Sciences. Springer, Berlin; 1973.
Samko SG, Kilbas AA, Marichev OI: Fractional Integrals and Derivatives: Theory and Applications. Gordon & Breach, Yverdon; 1993.
Rainville ED: Special Functions. Macmillan Co., New York; 1960.
Erdélyi A (Ed): Higher Transcendental Functions. McGraw-Hill, New York; 1953.
Kilbas AA, Saigo M, Trujillo JJ: On the generalized Wright function. Fract. Calc. Appl. Anal. 2004, 5: 437–460.
Virchenko N, Kalla SL, Al-Zamel A: Some results on a generalized hypergeometric function. Integral Transforms Spec. Funct. 2001, 12(1):89–100. 10.1080/10652460108819336
Virchenko N, Lisetska O, Kalla SL: On some fractional integral operators involving generalized Gauss hypergeometric functions. Appl. Appl. Math. 2010, 5(10):1418–1427.
Virchenko N, Rumiantseva OV: On the generalized associated Legendre functions. Fract. Calc. Appl. Anal. 2008, 11(2):175–185.
Rao SB, Patel AD, Prajapati JC, Shukla AK: Some properties of generalized hypergeometric function. Commun. Korean Math. Soc. 2013, 28(2):303–317. 10.4134/CKMS.2013.28.2.303
Rao SB, Shukla AK: Note on generalized hypergeometric function. Integral Transforms Spec. Funct. 2013, 24(11):896–904. 10.1080/10652469.2013.773327
Prajapati JC, Saxena RK, Jana RK, Shukla AK: Some results on Mittag-Leffler function operator. J. Inequal. Appl. 2013., 2013: Article ID 33
Rao SB, Salehbhai IA, Shukla AK: On sequence of functions containing generalized hypergeometric function. Math. Sci. Res. J. 2013, 17(4):98–110.
Rao SB, Prajapati JC, Shukla AK: Wright type hypergeometric function and its properties. Adv. Pure Math. 2013, 3(3):335–342. 10.4236/apm.2013.33048
Shukla AK, Prajapati JC: On a generalized Mittag-Leffler type function and generated integral operator. Math. Sci. Res. J. 2008, 12(12):283–290.
Srivastava HM, Tomovski Z: Fractional calculus with an integral operator containing a generalized Mittag-Leffler functions in the kernel. Appl. Math. Comput. 2009, 211: 198–210. 10.1016/j.amc.2009.01.055
Kilbas AA, Srivastava HM, Trujillo JJ North-Holland Mathematical Studies 204. In A Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam; 2006.
Hilfer R (Ed): Application of Fractional Calculus in Physics. World Scientific, Singapore; 2000.
Miller KS, Ross B: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York; 1993.
Mathai AM, Haubold HJ: Special Functions for Applied Scientists. Springer, Berlin; 2010.
Nyamoradi N, Baleanu D, Agarwal RP: Existence and uniqueness of positive solutions to fractional boundary value problems with nonlinear boundary conditions. Adv. Differ. Equ. 2013., 2013: Article ID 266
Baleanu D, Agarwal RP, Mohammadi H: Some existence results for a nonlinear fractional differential equation on partially ordered Banach spaces. Bound. Value Probl. 2013., 2013: Article ID 112
Baleanu D, Mustafa OG, Agarwal RP:Asymptotic integration of -order fractional differential equations. Comput. Math. Appl. 2011, 62(3):1492–1500. 10.1016/j.camwa.2011.03.021
Srivastava HM, Manocha HL: A Treatise on Generating Functions. John Wiley/Ellis Horwood, New York/Chichester; 1984.
Kilbas AA: Fractional calculus of the generalized Wright function. Fract. Calc. Appl. Anal. 2005, 8(2):113–126.
Gehlot KS, Prajapati JC: Fractional calculus of generalized k -Wright function. J. Fract. Calc. Appl. 2013, 4(2):283–289.
We are very grateful to the anonymous referees for their careful reading and helpful comments.
The authors declare that they have no competing interests.
All authors contributed equally to the manuscript. All authors read and approved the final manuscript.
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Rao, S.B., Prajapati, J.C., Patel, A.D. et al. Some properties of Wright-type generalized hypergeometric function via fractional calculus. Adv Differ Equ 2014, 119 (2014). https://doi.org/10.1186/1687-1847-2014-119
- fractional integral and differential operators
- generalized hypergeometric function
- Lebesgue measurable functions