Abstract
The aim of this present paper is to obtain a general expansion theorem involving H-functions of several complex variables. This is done by making use of a Taylor-like expansion in terms of a rational function obtained by means of fractional derivatives given recently by the authors. Special cases are also computed.
1. Introduction
In 1971, Osler obtained with the use of Cauchy integral formula for the fractional derivatives the following generalization for the Taylor's series [1]: where is a positive real number, , , and are arbitrary complex numbers, is an analytic function in a simply connected region and with is a regular and univalent function without zero in , and , being the largest integer not greater then . If and , then and the formula (1) reduces to This last formula is usually called the Taylor-Riemann formula and has been studied in several papers [2–6]. But none considered a more general expansion of in terms of a power series of an arbitrary quadratic, cubic, or higher degrees functions. Recently, the authors [7] obtained the power series of an analytic function in terms of the rational expression where and are two arbitrary points inside the region of analyticity of . In particular, we obtain the following expansion: Several restrictions are imposed on the functions and parameters in (3). The following list is considered. (i), , and are arbitrary complex numbers. (ii) is a real and is the integral index of summation. (iii), are fixed points in the -plane and , where , defines a double-loop curve on which the series (3) converges with . (iv) is on the loop around the point but as shown in Figure 1.
The aim of this paper is to obtain a new expansion theorem involving the -function of complex variables defined by Srivastava and Panda [8–11]. We will define and represent it in the following form [12, page 251, equation ()]: where , for all and Here, for convenience, abbreviates the -member array while abbreviates the array of pairs of parameters: and so on. Suppose, as usual, that the parameters: are complex numbers and the associated coefficients are positive real numbers such that where the integers , and are constrained by the inequalities ,, , and and the equality in (11) holds true for suitably restricted values of the complex variables .
The multiple Mellin-Barnes contour integral [12, page 251, equation ()] representing the multivariable -function (4) converges absolutely, under the conditions (12), when the points and various exceptional parameter values being tacitly excluded. Furthermore, we have (cf. [9, page 131, equation (1.9)]): where provided that each of the inequalities in (11)–(13) holds true.
Note that throughout this work, we will assume that the convergence and existence conditions corresponding appropriately to the ones detailed above are satisfied by each of the various -functions involved.
2. Pochhammer Contour Integral Representation for Fractional Derivative
The use of contour of integration in the complex plane provides a very powerful tool in both classical and fractional calculus. The most familiar representation for fractional derivative of order of is the Riemann-Liouville integral [13–15], 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 to modify the restriction to [15]. Another used representation for the fractional derivative is the one based on the Cauchy integral formula widely used by Osler [4, 16–18]. These two representations have been used in many interesting research papers. It appears that the less restrictive representation of fractional derivative according to parameters is the Pochhammer's contour definition introduced in [19, 20].
Definition 1. 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 , we define the fractional derivative of order of with respect to by For noninteger 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 2. denotes the principal value of the integrand in (19) at the beginning and ending point of the Pochhammer contour which is closed on Riemann surface of the multiple-valued function .
Remark 2. In Definition 1, the function must be analytic at . However it is interesting to note here that we could also allow to have an essential singularity at , and (19) would still be valid.
Remark 3. The Pochhammer contour never crosses the singularities at and in (19), 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 (19). However, integrating by parts times the integral in (19) by two different ways, we can show that , and are removable singularities (see [19]).
In 1985, Srivastava and Goyal [21, page 644, equation (17)] obtained the following fractional derivative formula, by using the well-known Riemann-Liouville's definition of the fractional derivative, for the multivariable -function (4): provided (in addition to the usual convergence and existence conditions) that , and where are given by (15).
Remark 4. Adopting the representation based on the Pochhammer contour for fractional derivative and in view of Remark 3, the restrictions can be modified to not a negative integer and to not a negative integer.
Letting and in (20) yields the following useful particular case:
3. Expansion Theorem and Special Cases
In this section, we establish the new expansion theorem for the -functions of several complex variables. Two special cases (presumably new) are also computed to demonstrate the importance of this new expansion theorem to the theory of special functions of mathematical physics.
Theorem 5. Assume that the convergence and existence conditions corresponding appropriately to the ones detailed in Section 1 are satisfied for each -functions involved. Assume also that the list of restrictions of (3) is satisfied, then the following expansionholds true.
Proof. By setting , in (3) and with the help of (22), we find first thatSubstituting the last result into (3) yields the desired result.
Example 6. For and the multivariable -function in the L.H.S. of (23) breaks up into product of -functions and there holds the following result [22]: Now setting , , , substituting by , into (25), replacing by , and making use of the following special case [23] gives Moreover, we also know from [21, page 649, equation (3.6)] that where denotes the Lauricella's hypergeometric function of variables [24, page 33, equation (4)]. Thus, we have with . Combining (27) and (29) with Theorem 5 yields the following new expansion involving the Lauricella's hypergeometric function of variableswith .
Example 7. Setting and , in (23) and with the help of the result given by Srivastava and Panda [10] connecting the generalized Lauricella function and the multivariable -function: we obtain the following expansion involving an extension of the Kampé de Fériet series [24, page 38, equation (24)] If we put , , , and in (32), we haveMaking the following changes and using the hypergeometric representation for the Jacobi functions [25, page 35] yields