Differentiation formulas of some hypergeometric functions with respect to all parameters

A family of the incomplete hypergeometric functions and associated integral transform and fractional derivative formulas

“…In the definitions (5), (6), (10), (11), (12) and (13), and throughout our present investigation, the argument x 0 is independent of the argument z ∈ C which occurs in the definitions (1), (10) and (11) and also elsewhere in this paper (see [35]). Moreover, as already pointed out by Srivastava et al [27,p.…”

“…In recent years, integral transforms involving fractional integral and fractional derivative formulas and various classes of special functions were investigated by many authors (see, for example, [2], [12], [22] and [23]). In the present sequel to the aforementioned recent work [25], by using essentially the same techniques as those that are detailed by Srivastava et al [25], we establish several (presumably new) integral transform and fractional derivative formulas involving the generalized incomplete hypergeometric functions p γ q and p Γ q given by the equations (10) and (11), respectively.…”

“…The function Γ(z), and its incomplete versions γ(s, x) and Γ(s, x), are known to play important rôles in the study of the analytic solutions of a variety of problems in diverse areas of science and engineering (see, for example, [1], [5], [18] and [29]; see also the recent papers [10], [23], [24], [26], [27], [16], [17], [34], [36], [37] and [38]).…”

“…the precise (sufficient) conditions under which the infinite series in the definitions (10) and (11) would converge absolutely can be derived from those that are well-documented in the case of the generalized hypergeometric function p F q (p, q ∈ N 0 ) (see, for details, [20, pp. 72-73] and [28, p. 20]).…”

“…Indeed, in their special case when x = 0, p Γ q (p, q ∈ N 0 ) would reduce immediately to the extensively-investigated generalized hypergeometric function p F q (p, q ∈ N 0 ). Furthermore, as an immediate consequence of the definitions (10) and 11, we have the following decomposition formula (see, for details, [27]):…”