Research Article | Open Access
Summation Formulas Obtained by Means of the Generalized Chain Rule for Fractional Derivatives
In 1970, several interesting new summation formulas were obtained by using a generalized chain rule for fractional derivatives. The main object of this paper is to obtain a presumably new general formula. Many special cases involving special functions of mathematical physics such as the generalized hypergeometric functions, the Appell function, and the Lauricella functions of several variables are given.
The fractional derivative of arbitrary order , , is an extension of the familiar th derivative of the function with respect to to nonintegral values of and is denoted by . The aim of this concept is to generalize classical results of the th order derivative to fractional order. Most of the properties of the classical calculus have been expanded to fractional calculus, for instance, the composition rule , the Leibniz rule [2, 3], the chain rule , and Taylor’s and Laurent’s series [5–7]. Fractional calculus also provides tools that make it easier to deal with special functions of mathematical physics .
The most familiar representation for fractional derivative of order of is the Riemann-Liouville integral ; that is, which is valid for and where the integration is done along a straight line from to in the -plane. By integrating by part times, we obtain This allows modifying the restriction to .
In 1970, Osler  introduced a more general definition of the fractional derivative of a function with respect to another function based on Cauchy’s integral formula.
Definition 1. Let be analytic in the simply connected region . Let be regular and univalent on and let be an interior or boundary point of . Assume also that for any simple closed contour in through . Then if is not a negative integer and is in , the fractional derivative of order of with respect to is defined by For nonintegral , the integrand has a branch line which begins at and passes through . The notation on this integral implies that the contour of integration starts at , encloses once in the positive sense, and returns to without cutting the branch line.
With the use of that representation based on the Cauchy integral formula for the fractional derivatives, Osler gave a generalization of the following result [11, page 19] involving the derivative of order of the composite function : where In particular, he found the following formula : where the notation means the fractional derivative of order of with respect to . Osler proved the generalized chain rule by applying the generalized Leibniz rule  for fractional derivatives to an important fundamental relation involving fractional derivatives discovered also by Osler [4, page 290, Theorem 2]. The fundamental relation which is the central point of this paper is given by the next theorem.
Theorem 2. Let and be defined and analytic on the simply connected region and let the origin be an interior or boundary point of . Suppose also that and are regular univalent functions on and that . Let vanish over simple closed contour in through the origin. Then the following relation holds true:
This fundamental relation is very useful to obtain very easily known and new summation formulas involving special functions of mathematical physics. For example, set , , and in (7). One sees easily that . Thus, one has The left-hand side is evaluated by using the well-known formula [12, page 83, Equation ] after replacing by . Expanding in power series, using (9), and replacing by after operation, one obtains Kummer’s summation formula where denotes the Gauss hypergeometric function  and holds for the Pochhammer symbol defined, in terms of the Gamma function, by
In this paper, we present several new summation formulas involving special functions of mathematical physics obtained by using the fundamental relation (7). In Section 2, we introduce the Pochhammer based representation for fractional derivatives and we recall a well-poised fractional calculus operator given by Tremblay . This well-poised operator will be used, throughout this paper, in order to ease the computations of fractional derivatives. Finally, Section 3 is devoted to the presentation of the main results. Many presumably new summation formulas are also given as special cases.
2. Pochhammer Contour Integral Representation for Fractional Derivative and the Well-Poised Fractional Calculus Operator
Definition 3. Let be analytic in a simply connected region . Let be regular and univalent on and let be an interior point of then if is not a negative integer, is not an integer, and is in , one defines the fractional derivative of order of with respect to by For nonintegers and , the functions and in the integrand have two branch lines which begin, respectively, at and , and both pass through the point without crossing the Pochhammer contour at any other point as shown in Figure 1. denotes the principal value of the integrand in (13) at the beginning and ending point of the Pochhammer contour which is closed on Riemann surface of the multiple-valued function .
Remark 5. The Pochhammer contour never crosses the singularities at and in (13); then we know that the integral is analytic for all and for all and for in . Indeed, the only possible singularities of are , and which can directly be identified from the coefficient of the integral (13). However, integrating by parts times the integral in (13) by two different ways, we can show that , and are removable singularities (see ).
Remark 6. A deep and complete study of the fundamental properties of the function with or , relative to the the values of the parameters , , and , can be found in .
In 1974, Tremblay  introduced the operator in order to deal with special functions more efficiently and to facilitate the obtention of new relations such as hypergeometric transformations. This operator will be useful in the proofs of the summation formulas given in the next section.
The operator is defined in terms of the fractional calculus operator as with .
This operator has many very useful properties. We chose to give only one of them which will be used in the proofs of the main results; that is, In terms of the fractional calculus operator , the modified fundamental relation (7) holds the following form: with .
It is worthy to mention that the operator has a lot more interesting properties and applications.
3. Main Results and Special Cases
In this section, we present a new general formula related to the generalized chain rule. We give many special cases involving special functions such as the first Appell function , the Lauricella function of several variables , and the generalized hypergeometric functions. These functions are evaluated most of the time at arguments related to the roots of unity.
Main Formula. Consider with and .
Example 7. Let , , and in (17). We, thus, have Expanding in power series, (20) becomes The right-hand side of (21) can be split in two parts Converting the terms involving Gamma function in the last expression into Pochhammer’s symbol and making some simplifications, we find Finally, rewriting the right-hand side of (23) in terms of generalized hypergeometric function and combining with (21), we obtain the following summation formula:
Special Cases with . Consider
Example 8. Setting and in (25) gives
where denotes the first Appell function . Replacing (27) in (26) yields the following summation formula:
It is interesting to note that setting and replacing by in (28), and using the well-known identity , reduces to the known summation formula 
Furthermore, with the help of the following reduction formulas for the Appell functions given by Nagel , in conjunction with (28) give, respectively, after simplifications the two summation formulas
Example 9. Let and let be an integer in (25). We have that The Lauricella function of variables is defined as [18, page 60] with . We can easily see that the following relation, holds true. So, using this last relation in conjunction with (33) provides the following summation formula:
Let us examine another special case of (17) when is not equal to 1.
Example 10. Set and in (25). We have Expanding in power series, we find for the left-hand side of (37) The right-hand side of (37) yields after expanding in power series and using property (15) Using (27) and making some elementary simplifications, the last formula reduces to Combining (38) and (40) and putting provide the following presumably new summation formula:
Cases with . Consider
Remark 12. The cases where with must be treated very carefully as .
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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