When the Upper Parameters Differ by Integers

In many problems arising in physical sciences and statistics, hypergeometric functions, in logarithmic cases, are applicable, which is known from the monograph of Mathai and Haubold [4] and Mathai [3], etc. A detailed account of such applications is available from the two monographs by Mathai and Saxena [5], [6]. Expansions of the Gauss’s hypergeometric functions in logarithmic cases are given in the monograph of Erdélyi et al. [1]. In a series of papers by Saigo et al. [9] [25], numerous properties have been investigated which exhibits the behaviour of generalized hypergeometric functions near the boundaries of their regions of convergence of series defining these functions. In deriving some of these results, the expansions of Gauss’s hypergeometric function in logarithmic cases given in the monograph by Magnus et al. [2] are used. The object of this article is to derive three expansions of a generalized hypergeometric function 4F3(·), when the upper parameters differ by integers. Though the results established in this article are special cases of a general continuation formula for pFq [7, Entry 78 in §7.2.3], yet they are sufficiently general in nature and unify a number of known results in literature [1, p.63], [27, pp.41-43], etc. Further the results are obtained in a neat and compact form which may be used in investigating the asymptotic behaviour of the generalized hypergeometric functions. From [8, p.101], we know that the generalized hypergeometric function 4F3(·) has the following Mellin-Barnes integral representation