GENERALIZED FRACTIONAL CALCULUS OF THE ALEPH-FUNCTION INVOLVING A GENERAL CLASS OF POLYNOMIALS?

R.K.SAXENAD.KUMAR

Department of Mathematics and Statistics，Jai Narain Vyas University，Jodhpur 342005，India

GENERALIZED FRACTIONAL CALCULUS OF THE ALEPH-FUNCTION INVOLVING A GENERAL CLASS OF POLYNOMIALS?

R.K.SAXENAD.KUMAR

Department of Mathematics and Statistics，Jai Narain Vyas University，Jodhpur 342005，India

E-mail：ram.saxena@yahoo.com；dinesh dino03@yahoo.com

The object of this article is to study and develop the generalized fractional calculus operators given by Saigo and Maeda in 1996.We establish generalized fractional calculus formulas involving the product of?-function，Appell function F3and a general class of polynomials.The results obtained provide unification and extension of the results given by Saxena et al.［13］，Srivastava and Grag［17］，Srivastava et al.［20］，and etc.The results are obtained in compact form and are useful in preparing some tables of operators of fractional calculus. On account of the general nature of the Saigo-Maeda operators，?-function，and a general class of polynomials a large number of new and known results involving Saigo fractional calculus operators and several special functions notably H-function，I-function，Mittag-Leffler function，generalized Wright hypergeometric function，generalized Bessel-Maitland function follow as special cases of our main findings.

generalized fractional calculus operators；a general class of polynomials；?-function；H-function；I-function；generalized Wright hypergeometric function；Mittag-Leffler function；generalized Bessel-Maitland function

2010 MR Subject Classification 26A33；33E20；33C45；33C60；33C70

1 Introduction and Preliminaries

Fractional calculus has gained importance and popularity during the last four decades or so due to its various applications in many branches of sciences and engineering.

The fractional integral operator involving various special functions，were found significant importance and applications in various sub-field of application mathematical analysis.Since last five decades，a number of workers like Love［3］，Srivastava and Saxena［19］，Debnath and Bhatta［1］，Saxena et al.［13-15］，Saigo［7］，Samko，Kilbas and Marichev［9］，Miller and Ross［5］，and Ram and Kumar［6］，etc.studied in depth，the properties，applications and different extensions of various hypergeometric operators of fractional integration.

These operators reduce to the fractional integral operators introduced by Saigo［7］，due to the following relations:

and

Let α，α′，β，β′，γ∈C，γ＞ 0 and x∈R+，then the generalized fractional differentiation operators［8］involving Appell function F3as a kernel can be defined as

These operators reduce to Saigo derivative operators［7，8］as

Further from（［8］，p.394，eqns.（4.18）and（4.19）），we also have

where Re（γ）＞0，Re（ρ）＞max［0，Re（α+α′+β-γ），Re（α′-β′）］，and

where Re（γ）＞0，Re（ρ）＜1+min［Re（-β），Re（α+α′-γ），Re（α+β′-γ）］.

Following Saxena&Pog′any［11，12］，we define the Aleph-function in terms of the Mellin-Barnes type integrals as follows:

The integration path L=Lωγ∞，γ∈? extends from γ-ω∞to γ+ω∞，and is such that the poles of Γ（1-aj-Ajξ），j=（the symbol 1，n is used for 1，2，···，n）do not coincide with the poles of Γ（bj+Bjξ），j=The parameters pi，qiare non-negative integers satisfying the condition 0≤n≤pi，1≤m≤qi，τi＞0 for i=The parameters Aj，Bj，Aji，Bji＞0 and aj，bj，aji，bji∈C.An empty product in（1.14）is interpreted as unity.The existence conditions for the defining integral（1.13）are given below

where

Remark 1.1 For τi=1，i=1，r，in（1.13）we get the I-function due to Saxena［10］，defined in the following manner

Remark 1.2 If we set r=1，then（1.19）reduces to the familiar H-function［4］

Recently，generalized fractional calculus formulae of the Aleph-function associated with the Appell function F3was given by Saxena et al.［14，15］，and Ram&Kumar［6］.

Also，Smn［x］occurring in the sequel denotes the general class of polynomials introduced by Srivastava and studied by Srivastava and Garg［17］，Srivastava et al.［20］:where m is an arbitrary positive integer and the coefficient An，k（n，k≥0）are arbitrary constants，real or complex.On suitably specialize the coefficients An，k，Smn［x］yields a number of known polynomials as its special cases.

2 Fractional Integral Formulas

In this section we will establish two fractional integration formulae for?-function（1.13）and a general class of polynomials defined by（1.21）.The conditions as given in（1.15）-（1.18）hold true.

Proof In order to prove（2.1），we first express a general class of polynomials occurring on its left-hand side in the series from（1.21），replacing the?-function in terms of Mellin-Barnes contour integral with the help of equation（1.13），interchange the order of summations，we obtain the following form（say I）:Finally，reinterpreting the Mellin-Barnes counter integral in terms of the?-function，we obtain the right-hand side of（2.1）.This completes the proof of Theorem 2.1. □

In view of relation（1.3），we arrive at the following corollary concerning Saigo fractional integral operator［7］.

where the conditions of existence of the above corollary follows easily with the help of（2.1）.

Remark 2.3 We can also obtain results concerning Riemann-Liouville and Erd′elyi-Kober fractional integral operators by putting β=-α and β=0 respectively in Corollary 2.2.

Theorem 2.4 Let α，α′，β，β′，γ，z，ρ∈C，Re（γ）＞0，μ＞0，λj＞0（j∈｛1，2，···，s｝），Re（ρ）-μ＜1+min［Re（-β），Re（α+α′-γ），Re（α+β′-γ）］.Further，letthe constants aj，bj，aji，bji∈C，Aj，Bj，Aji，Bji∈R+（i=1，···，pi；j=1，···，qi），τi＞0 for i=Then the following relation holds

Proof In order to prove（2.3），we first express a general class of polynomials occurring on its left-hand side in the series from（1.21），replacing the?-function in terms of Mellin-Barnes contour integral with the help of equation（1.13），interchange the order of summations，we obtain the following form（say I）:

Finally，reinterpreting the Mellin-Barnes counter integral in terms of the?-function，we obtain the right-hand side of（2.3）.This completes the proof of Theorem 2.4. □

In view of relation（1.4），we arrive at the following corollary concerning Saigo fractional integral operator which is believed to be new.

where the conditions of existence of the above corollary follows easily from Theorem 2.4.

Remark 2.6 We can also obtain results concerning Riemann-Liouville and Erd′elyi-Kober fractional integral operators by putting β=-α and β=0 respectively in Corollary 2.5.

3 Fractional Derivative Formulas

In this section we will establish two fractional derivative formulae for?-function（1.13）and a general class of polynomials defined by（1.21）.The conditions as given in（1.15）-（1.18）hold true.

Proof In order to prove（3.1），we first express a general class of polynomials occurring on its left-hand side in the series from（1.21），replacing the?-function in terms of Mellin-Barnes contour integral with the help of equation（1.13），interchange the order of summations，we obtain the following form（say I）:

here k=［-Re（γ）］+1，and by usingwhere m ≥k in the above expression，and re-interpreting the Mellin-Barnes counter integral in terms of the?-function， we obtain the right-hand side of（3.1）.This is complete proof of Theorem 3.1.

In view of relation（1.9），then we arrive at the following corollary concerning Saigo fractional derivative operator which is also believed to be new.

where the conditions of existence of the above corollary follow easily with the help of（3.1）.

Remark 3.3 We can also obtain results concerning Riemann-Liouville and Erd′elyi-Kober fractional derivative operators by putting β=-α and β=0 respectively in Corollary 3.2.

Proof In order to prove（3.3），we first express a general class of polynomials occurring on its left-hand side in the series from（1.21），replacing the?-function in terms of Mellin-Barnes contour integral with the help of equation（1.13），interchange the order of summations，we obtain the following form（say I）:here k=［Re（γ）］+1，and by using，where m ≥k in the above expression，and reinterpreting the Mellin-Barnes counter integral in terms of the?-function，we obtain the right-hand side of（3.3）.This is complete proof of Theorem 3.4.

In view of relation（1.10），we arrive at the following corollary concerning Saigo fractional derivative operator［7］which is also believed to be new.

where，the conditions of existence of the above corollary follow easily from Theorem 3.4.

Remark 3.6 We can also obtain results concerning Riemann-Liouville and Erd′elyi-Kober fractional derivative operators by putting β=-α and β=0 respectively in Corollary 3.5.

4 Special Cases and Applications

This section deals with certain interesting special cases of Theorem 2.1.We can easily obtain similar results for Theorem 2.4，Theorem 3.1，Theorem 3.4 but we are not presenting them due to lack of space.

（i）If we put τi=1，i=1，r in Theorem 2.1 and take（1.19）into account，then Aleph function reduces to the I-function［10］and we arrive at the following result:

（ii）Now，if we put τi=1，i=1，r and set r=1 in Theorem 2.1 and take（1.20）into account，then Aleph function reduces to the H-function［4］and we arrive at the following result:

（iv）If we use the relation with Wright generalized hypergeometric functionpψq（［4］，eqn.（1.140），p.25）in（4.2）then the following result is obtained

（v）If we use the following relation in（4.2），given by Saxena et al.（［4］，eqn.（1.127），p. 23），

where Jv（z）is the ordinary Bessel function of the first kind；then we arrive at

Remark 4.1 We can also obtain results for modified Bessel functions Kv（z）and Yv（z）by following similar lines as done in（4.5），where Kv（z）is the modified Bessel function of the third kind or Macdonald function and Yv（z）is the modified Bessel function of the second kind or the Neumann function.

（vi）Further，if we use the following relation in（4.2），given by Saxena et al.（［4］，eqn.（1.139），p.25），

（vii）If we use the relation with Kummer's confluent hypergeometric functions φ（a；d；-z）（［4］，eqn.（1.130），p.24），in（4.2）then we obtain the following result:

Remark 4.2 Similar results can also be obtained for Gauss'hypergeometric function2F1（b，a；d；-ztμ）by using the following relation:

（viii）In（4.2），by using the relation connecting H-function and MacRobert's E-function

we obtain the following result:

（ix）By using the relation connecting Whittaker function and the H-function

then we obtain one more special case of（4.2）give as follow:

（x）If we set a general class of polynomialsto unity，and reduce the?-function to Fox's H-function，then we can easily obtain the known results given by Saxena and Saigo［16］.

5 Conclusion

In the present paper，we have given the four theorems of generalized fractional integral and derivative operators given by Saigo-Maeda.The theorems have been developed in terms of the product of?-function and a general class of polynomials in a compact and elegant form with the help of Saigo-Maeda power function formulae.Most of the given results have been put in a compact form，avoiding the occurrence of infinite series and thus making them useful in applications.

Acknowledgments The second author would like to thank NBHM Department of Atomic Energy，Government of India，Mumbai for the finanicai assistance under PDF sanction no.2/ 40（37）/2014/R&D-II/14131.

［1］Debnath L，Bhatta D.Integral Transforms and Their Applications.Boca Raton FL:Chapman and Hall/CRC Press，2006

［2］Erd′elyi A，Magnus W，Oberhettinger F，Tricomi F G.Higher Transcendental Function.Vol I.New York，Toronto，London:McGraw-Hill，Reprinted:Krieger，Melbourne-Florida，1953

［3］Love E R.Some integral equations involving hypergeometric functions.Proc Edin Math Soc，1967，15（3）: 169-198.

［4］Mathai A M，Saxena R K，Haubold H J.The H-function:Theory and Applications.New York:Springer，2010

［5］Miller K S，Ross B.An introduction to Fractional Calculus and Fractional Differential Equations.New York:John Wiley and Sons Inc，1993

［6］Ram J，Kumar D.Generalized fractional integration of the?-function.J Rajasthan Acad Phy Sci，2011，10（4）:373-382

［7］Saigo M.A remark on integral operators involving the Gauss hypergeometric functions.Math Rep，College General Ed Kyushu Univ，1978，11:135-143

［8］Saigo M，Maeda N.More generalization of fractional calculus//Transform Methods and Special Functions. Varna，Bulgaria，1996:386-400

［9］Samko S G，Kilbas A A，Marichev O I.Fractional Integrals and Derivatives:Theory and Applications. Yverdon:Gordon and Breach，1993

［10］Saxena V P.Formal solution of certain new pair of dual integral equations involving H-functions.Proc Nat Acad Sci India Sect，1982，51A:366-375

［11］Saxena R K，Pog′any T K.On fractional integration formulae for Aleph functions.Appl Math Comput，2011，218:985-990

［12］Saxena R K，Pog′any T K.Mathieu-type Series for the?-function occurring in Fokker-Planck equation.Eur J Pure Appl Math，2010，3（6）:980-988

［13］Saxena R K，Ram J，Chandak S，Kalla S L.Unified fractional integral formulae for the Fox-Wright generalized hypergeometric function.Kuwait J Sci Eng，2008，35A（1）:1-20

［14］Saxena R K，Ram J，Kumar D.Generalized fractional differentiation for Saigo operators involving Aleph-Function.J Indian Acad Math，2012，34（1）:109-115

［15］Saxena R K，Ram J，Kumar D.Generalized fractional differentiation of the Aleph-Function associated with the Appell function F3.Acta Ciencia Indica，2012，38（4）:781-792

［16］Saxena R K，Saigo M.Generalized fractional calculus of the H-function associated with the Appell function F3.J Fract Calc，2001，19:89-104

［17］Srivastava H M，Garg M.Some integrals involving a general class of polynomials and the multivariable H-function.Revista，Roumaine Physics，1987，32:685-692

［18］Srivastava H M，Owa S.An application of the fractional derivative.Math Japon，1984，29:383-389

［19］Srivastava H M，Saxena R K.Operators of fractional integration and their applications.Appl Math Comput，2001，118:1-52

［20］Srivastava H M，Saxena R K，Ram J.Some multidimensional fractional integral operations involving a general class of Polynomials.J Math Anal Appl，1995，193:373-389

?Received December 30，2013；revised March 18，2015.