problems and historical background

In an attempt to make the book self-contained, this chapter aims at providing the necessary background material in the theory of hypergeometric series in one and more variables. We begin by introducing the reader to two interesting problems in the study of multiple Gaussian hypergeometric series. We also present a brief discussion of some possible applications of many of the multiple hypergeometric series considered in this book.
1.1 PROBLEMS AND HISTORICAL BACKGROUND
Hypergeometric series which reduce to the familiar Gaussian series
(1)

whenever only one variable is non-zero may be called multiple Gaussian hypergeometric series. Two interesting problems in the theory of multiple Gaussian hypergeometric series consist in constructing all distinct series and in establishing their regions of convergence. Both of these problems are rather straightforward for single series, and they have been completely solved in the case of double series [see, for example, Erdelyi et al. (1953, Section 5.7)]. This book aims at presenting a systematic discussion of the complexity of the problems when the dimension exceeds two; indeed, we shall give the complete solution of each of the problems in case of the triple Gaussian hypergeometric series.

We begin by briefly summarizing the various Gaussian series hitherto introduced. Of course, the only single Gaussian series is F(a, b; c; z) defined by (1). Fourteen distinct double Gaussian series exist: Appell (1880) introduced F₁,...,F₄, but the set was not completed until after Horn (1931) gave the remaining ten series (G₁, G₂, G₃, H₁,...,H₇).
Lauricella (1893) introduced 14 triple Gaussian series (F₁,...,F₁₄). More precisely, he defined four n-dimensional series and (which, for n = 3, correspond to the triple series F₁ , F₂, F₅ and F₉, respectively), and ten further triple series. This latter set of ten triple series F₃, F₄, F₆, F₇, F₈, F₁₀,...,F₁₄ apparently fell into oblivion, except that there is an isolated appearance of F₈ in Mayr (1932) who came across this triple series while evaluating certain infinite integrals. Saran (1954a) initiated a systematic study of these ten triple Gaussian series of Lauricella’s set, and his notations F_E, E_F,..._tF_T now prevail in the literature. Since then a few additional triple Gaussian series have been introduced; for example, G_A and G_B by Pandey (1963a), H_A, H_B and H_c by Srivastava (1964a, 1967b), Gc by Srivastava (1972a), and so on [cf. Exton (1982a)]. Yet only about 50 triple Gaussian series have appeared in the literature. In Chapter 3 we shall construct the entire set of 205 distinct triple Gaussian hypergeometric series.
Special quadruple Gaussian series have been considered by Exton (1972a, 1973a), and Karlsson (1976a) has studied some even-dimensional Gaussian series. Certain Gaussian series in n variables have been introduced by Erddlyi (1939a) and by Exton (1973b, 1976b).
We remark in passing that, although Appell (1880) deserves to be credited with the first systematic study of multiple hypergeometric series, some instances of such series did appear before 1880. Polynomials that are, in fact, particular cases of Appell’s double series F₃ were introduced by Hermite (1865), and their generalization to n variables (that is, a particular case of Lauricella’s series was studied by Didon (1868). Even more remarkable is the fact that Schlafli (1874) considered the series and gave its single Eulerian integral representation which (for n=2) is usually attributed to Picardf (1880). The subject of Schlafli’s paper being conformal representation, it apparently remained unnoticed as far as the theory of hypergeometric series is concerned.
The problem of convergence hardly exists for single hypergeometric series. The region of convergence of the Gaussian series (1) is obtained immediately by appealing to d’Alembert’s ratio test. A generalization of this test was given by Horn (1889) whose theorem on the convergence of double (and triple) hypergeometric series will be considered in greater detail in Section 2.2 (and in Section 5.1). Of course, Horn’s theorem applies readily to the 14 double Gaussian series [cf. Erddlyi et al. (1953, Section 5.7)], but occasionally regions of convergence are established by applying Stirling’s theorem; see, for instance, Appell and Kampd de Fdriet


Sec. 1.2] Gaussian Hypergeometric Series and Its Generalizations

(1926, Section 3), Bailey (1935a, Section 9.1), and Slater (1966, Section 8.1.1).

Regions of convergence have usually been given together with the triple Gaussian series appearing in the literature. Although allegedly derived from Horn’s theorem, however, many of these regions are incorrectly stated, and corrections were given by Karlsson (1974a). The present investigation is a continuation of Karlsson (1974a); it is concerned chiefly with the establishment of the regions of convergence for the distinct series constructed. In a number of cases we shall find that methods based upon Stirling’s theorem are advantageous, as pointed out by Karlsson (1974a). Indeed such methods were applied to certain series in n variables [Karlsson (1978a)] and to the aforementioned even-dimensional series [Karlsson (1976a)].
In addition to the Gaussian series, which have received the greatest attention in the literature, confluent series! have been considered. The above-mentioned problems have been fully solved for double confluent series. Twenty distinct series exist; seven were introduced by Humbert (1920-21), and the remaining ones by Horn (1931) and by Borngasser (1933). Certain triple confluent series were considered by Jain (1966a) and by Exton (1970), but the entire set has not been given. We restrict ourselves to a suitable subset of triple confluent series, which will be given as a preparation to the construction of the triple Gaussian series.
We shall not, in general, be concerned here with the above-mentioned problems involving multiple hypergeometric series of superior (or general) order. Such series have been introduced from time to time [see Kampe de F6riet (1921), Burchnall and Chaundy (1941), Srivastava (1967a), Srivastava and Daoust (1969a, b), and Srivastava (1970d)]. While a brief account of such multiple hypergeometric series is presented in Exton (1976b), we shall include their definitions and important special cases in Sections 1.3,1.4 and 1.5.