On Applications of Fractional Calculus Involving Summations of Series

A significantly large number of earlier works on the subject of fractional calculus give interesting account of the theory and applications of fractional calculus operators in many different areas of mathematical analysis (such as ordinary and partial differential equations, integral equations, special functions, summation of series, et cetera). The main object of the present paper is to obtain number of summat ions of series concerning generalized hypergeometric functions. Our finding provides interesting unifications and extensions of a number of new and known results.


Introduction
One of the most frequently encountered tools in the theory of fractional calcu lus (that is, differentiat ion and integration of an arbitrary real or co mplex order) is furn ished by the familiar differintegral operator defined and represented by Oldham and Spanier [12]: (1. 1) and (1.2) where n is the least positive integer such that n>q.
provides a generalization of the familiar differential and integral operator, v iz., and }.
For a=0 the operator is given by (1.3) corresponding essentially to the classical Riemann-Liouville fractional derivative (or integral) of order (or -). Moreover, when , Equation (1.1) may be identified with the definition of the familiar Weyl fract ional derivative (or integral) o f order (or -). In recent years there has appeared a great deal of literature discussing the application of the aforementioned fractional calculus operators in a number of areas of mathematical analysis (such as ordinary and partial differential equations, integral equations, summation of series, et cetera) and now stands on fairly firm footing through the research contribution of various authors (cf., e.g., [2], [5][6][7], [9][10][11][12][13][14], [16] and [17]). In the present paper main object is to obtain number of summations of series concerning generalized hypergeometric functions.
The familiar Leibniz rule for ord inary derivatives admits itself of the fo llo wing extension in terms of the Rie mann-Liouville operator defined by (1.3): The generalized Leibniz rule (1.4), wh ich was also applied earlier by Galué et al. [5] o rder to derive the summation identity: Suffers fro m an apparent drawback in the sence that the interchange of the function u(z) and v(z) on the right-hand side is not obvious. (see also Galué et al. for several summation formu las [6] contained in the Chen-Srivastava [2] ) which she deduced by suitable specializing the function u(z) and v(z) in the summation identity (1.5) above.) A further symmetrical generalized of (1.4) considered by Watanabe [17] and Osler [13], without such a drawback, is given by (cf., e.g., Samko  The condition of valid ity of the above results is given by T. J. Osler [13, p. 664-665]).
The generalized hypergeometric function of one variable viz., defined and represented as follows (see e.g. [15, p.19]) is also required here: The Laguerre polynomials defined and represented as follows (see e.g.[1, p.775]): where is the Pochhammer sy mbol and is a confluent hypergeometric function of the first kind (see e.g. [8]).

Main results
In this section, we shall establish some new summation formu lae for the generalized hypergeometric function .
Summati on Formulae 2.1 The conditions of validity of the above results follow easily fro m the conditions given by T. J. Osler [ The conditions of validity of the above results follow easily fro m the conditions given by T. J. Osler [ provided that The conditions of validity of the above results follow easily fro m the conditions given by T. J. Osler [ 13, p. 664-665] The conditions of validity of the above results follow easily fro m the conditions given by T. J. Osler [ 13, p. 664-665]).

Proofs:
The results are obtained by assigning particular values to the functions u (z) and v (z) in the generalized Leibniz ru le (1.6).  Similarly, if we take in (1.6), we easily arrive at the required formu lae (2.5) after a little simp lificat ion on making use of similar lines of proof as adopted in (2.1) and using known results [4, p.190

Special Cases
In view of the large number of parameters involved in the summat ions of series established above, these summations of series are capable of yielding a number of known and new results. We record here only one special case for lack of space. For example: If, we take in (2.1) and making use of the following well-known result on both the sides of the resulting result of (2.1) (cf., e.g., Erdély i et al. [4, p.185