Acta mathematica scientia, Series B >
GENERALIZED FRACTIONAL CALCULUS OF THE ALEPH-FUNCTION INVOLVING A GENERAL CLASS OF POLYNOMIALS
Received date: 2013-12-30
Revised date: 2015-03-18
Online published: 2015-09-01
Supported by
The second author would like to thank NBHMDepartment of Atomic Energy, Government of India, Mumbai for the finanicai assistance under PDF sanction no.2/40(37)/2014/R&D-II/14131.
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 N-function, Appell function F3 and 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, N-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.
R. K. SAXENA , D. KUMAR . GENERALIZED FRACTIONAL CALCULUS OF THE ALEPH-FUNCTION INVOLVING A GENERAL CLASS OF POLYNOMIALS[J]. Acta mathematica scientia, Series B, 2015 , 35(5) : 1095 -1110 . DOI: 10.1016/S0252-9602(15)30042-4
[1] Debnath L, Bhatta D. Integral Transforms and Their Applications. Boca Raton FL:Chapman and Hall/CRC Press, 2006
[2] Erdélyi 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, Pogány T K. On fractional integration formulae for Aleph functions. Appl Math Comput, 2011, 218:985-990
[12] Saxena R K, Pogány 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 gener-alized 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
/
| 〈 |
|
〉 |