- January 27, 2018
- Posted by: RSIS
- Categories: Applied Science, Engineering, Mathematics

**Submission Deadline-13th August 2024**

**Submission Deadline-20th August 2024**

**Submission Deadline-20th August 2024**

**International Journal of Research and Scientific Innovation (IJRSI) | Volume IV, Issue XII, December 2017 | ISSN 2321–2705 **

**Some Results Conncered to the I-Function of Fractional Calculus**

**V. S. Dhakar[1] and Smita Sharma[2]**

**[1, 2]Department of Mathematics, ITM Group of Institutions Gwalior, Gwalior-474001, INDIA**

Abstract:- The main objects of this paper is to derive the results for the I- function involving the Riemann-Liouville, the Weyl and such other fractional calculus operators as those based on the Cauchy- Goursat integral formula. The results derived in this paper are basic in nature and may include a number of known and new results as special cases.

Keyward: – Riemann-Liouville, the Weyl Oprators, H-function, G-function, and I function

I. INTRODUCTION AND PRELIMINARIES

In view of the generality of the I-function, on specializing the various parameters, we can obtain from our results, several results involving a remarkably wide variety of useful functions, which are expressible in terms of H-function, Gfunction, Fox’s Wright function, generalized mittag-Leffler functions and their various special cases. Thus, the results presented in this paper would at once yield a very large number of results involving a large variety of special functions occurring in the problems of science, engineering, mathematical physics etc. In 1961, Charles Fox [3] introduced a function which is more general in than the Meijer’s G-function and this function is well known in the literature of special functions as Fox’s Hfunction.

REFERENCES [1]. Chaurasia, V.B.L. and Singh, J., “Fractional calculus results pertaining to special functions” Int. J. Contemp. Math. Sci. Vol. 5 No. 10 (2010) pp: 2381-2389 [2]. Erdelyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F. G., “Tables of integral transforms” Vol. II, MaGra-Hill Book Company, New York, (1954) [3]. Fox, C., “The G- and H-functions as symmetrical Fourier kernels” Trans. Amer. Math. Soc., 98(1961) pp: 395-429 [4]. Nishimoto, K., “An essence of Nishimoto’s fractional calculus (Calculus of the 21st century): Integrations and Differentiations of arbitrary order” Descartes Press, Koriyama, (1991) [5]. Nishimoto, K.,”Fractional calculus, Vols. I-V” Descartes Press, Koriyama, (1984, 1987, 1989, 1991, 1996) [6]. Podlubny, I., “Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, Some Methods of Their Solution and Some of Their Applications to Methods of Their Solution and Some of Their Applications” Academic Press, New York, NY, USA, (1999) [7]. Samko, S. G., Kilbas, A. A. and Marichev, O. I., “Integrals and derivatives of fractional order and some of their applications (Nauka I Tekhnika, Minsk, 1987) translated in fractional integrals and derivatives: Theory and applications” Gordon and Beach Science Publishers, Reading, (1993) [8]. Saxena, V. P.,”The I-function” Anamaya Publishers, New Delhi, (2008) [9]. Sharma, K. and Dhakar, V.S., “On Fractional Calculus and Certain Results Involving K2 – Function” GJSFR.Vo.11 Version 1.0 (2011) pp: 17-21. [10]. Shrivastava, H. M., Buschman, R. G., “Theory and Applications of Convolution integral equations, Koluwer series on mathematics and its applications 79” Koluwer Academic Publishers, Dordrecht, (1992) [11]. Shrivastava, H. M., Owa, S. and Nishimoto, K.,”Some fractional differintegral equations” J. Math. Anal. Appl. 106 (1995) pp: 360- 366.