An Expansion Theorem Involving H-Function of Several Complex Variables

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. Figure 1: Multiloops contour. 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