Generalized Functions for the Fractional Calculus and Dirichlet Averages 1

The objective of this article is to investigate the Dirichlet averages of the more generalized function for the fractional calculus. (in particular R-function introduced by Hartley and Lorenzo) Representation of such relations are obtained in terms of fractional integral operators in particular Reimann-liouville integrals. Some interesting special cases of the established results associated with generalized MittagLeffler functions due to Saxena, etal , Srivastava and Tomovski, are deduced. Mathematics Subject Classifications: 26A33 and 33C20. 1200 Farooq Ahmad, D. K. Jain and Renu Jain


INTRODUCTION
Extension of concept has been a good motivating factor for research and development in mathematical sciences.Examples are the extension of real number system to complex numbers, factorial of integers to gamma functions etc. Extension of integer order derivatives and integrals to arbitrary order.In literature, number of papers have used two important functions for the solution of fractional order differential equations, the well-known Mittag-leffler function ‫ܧ‬ ఊ ‫ݐܽ(‬ ) called "Queen function "of fractional calculus and the F-function ‫ܨ‬ ሺܽ, ‫)ݐ‬ of Hartley and Lorenzo(1998).As these functions provided direct solutions and important understanding for the fundamental liner fractional order differential equations and for the related initial value problem.But in this paper we are taken the generalized function for the fractional calculus called R-function, it is of significant usefulness to develop a generalized function when fractionally differ integrated (by any order) returns itself.Such a function would greatly ease the analysis of fractional order differential equation.To overcome this , the following function was proposed by Hartley and Lorenzo, called R-function defined as follows [5&6] The more compact notation

Generalized functions for fractional calculus 1201
Definitions and preliminaries used in the paper A typical combination of gamma functions is given a name, because it has simple and useful integral representation (2) If x and y are real and +ve , then normalized integrand of (2) occurs in statistics as the type -I beta density function .For complex values of x, y the integrand defines complex measure on the unit interval, as Where C> = {zϵC; z>0} And R> = {xϵR;x>0}.
Our paper is devoted to the study of the Dirichlet averages of the generalized function for the fractional calculus (R-Function) (1) in the form M , ൣሺa, cሻ; ൫ߚ, ߚ / ; x , y ൯൧= ‫‬ ܴ ,

Representation of and , in terms of Reimann-Liouville Fractional Integrals
In this section we deduced representations for the Dirichlet averages ܴ ሺߚ, ߚ | , ‫,ݔ‬ ‫ݕ‬ሻ and M , ሺߚ, ߚ | ; x, yሻwith fractional integral operators.
This proves the theorem.

Special Cases
In this section, we consider some particular cases of the above theorem by setting V = q-1, c= 0, we get Dirichlet averages of ML-function reported by setting y= 0 in above equations, we get well-known result reported in , ‫ݔ‬ .
In particular, when

CONCLUSION
This paper has presented a new function for the fractional calculus, it is called the Rfunction .The R-function is unique in that it contains all of the derivatives and integrals of the F-function.The R-function has the eigen -property, that is it returns itself on q th order differ-integration.Special cases of the R-function also include the exponential function, the sin , cosine , hyperbolic sine and hyperbolic Cosine functions.
The value of the R-function is clearly demonstrated in the dynamic thermocouple problem where it enables the analyst to directly inverse transform the Laplace domain solution , to obtain the time domain solution.
Dirichlet averages : A Dirichlet average of a function denotes a certain kind of integral average with respect to a Dirichlet measure .The concept of Dirichlet average was introduced by Carlson in 1977 and others [1] as follows F(b,z) = ‫‬ ‫݀‪ሻ‬ݖݑ‪݂ሺ‬‬ ఓ ್ ሺu).‫݁ݎ݄݁ݓ‬ ߤ is a Dirichlet measure on the standard simplex E in ܴ ିଵ and ‫ݖݑ‬ = ∑ ‫ݑ‬ ‫ݖ‬ ୀଵ , ‫ݑ‬ = 1-‫ݑ‬ ଵ െ ‫ݑ‬ ଶ … -‫ݑ‬ ିଵ .The quantity ‫ݖݑ‬ is both convex combination of ‫ݖ‬ ଵ , ‫ݖ‬ ଶ , ‫ݖ‬ ଷ , … ‫ݖ‬ and linear function of independent variables ‫ݑ‬ ଵ , ‫ݑ‬ ଶ , … ‫ݑ‬ ିଵ .