# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Generalized hypergeometric functions at unit argument. (English) Zbl 0754.33003

The behaviour of the Gaussian hypergeometric series (for $p=1$)

${}_{p+1}{F}_{p}\left(\genfrac{}{}{0pt}{}{{a}_{1},{a}_{2},\cdots ,{a}_{p+1}}{{b}_{1},\cdots ,{b}_{p}}|z\right)=\sum _{n=0}^{\infty }\frac{{\left({a}_{1}\right)}_{n}{\left({a}_{2}\right)}_{n}\cdots {\left({a}_{p+1}\right)}_{n}}{{\left({b}_{1}\right)}_{n}\cdots {\left({b}_{p}\right)}_{n}n!}{z}^{n},\phantom{\rule{2.em}{0ex}}|z|<1,$

near the unit argument is given by a known analytic continuation formula when $s={\sum }_{j=1}^{p}{b}_{j}-{\sum }_{j=1}^{p+1}{a}_{j}$ is not an integer. R. J. Evans and D. Stanton [SIAM J. Math. Anal. 15, 1010-1020 (1984; Zbl 0547.33001)] obtained continuation formulas near $z=1$ for ${}_{p+1}{F}_{p}$, $p=2$, when $s=0$. The author of this paper obtains a continuation formula near $z=1$ for $p=2$ and unrestricted $s$ and also for $p=3$ or 4 when $s$ is not an integer. M. Saigo and H. M. Srivastava [Proc. Am. Math. Soc. 110, No. 1, 71-76 (1990; Zbl 0706.33004)] obtained a continuation formula near $z=1$ for arbitrary $p$ when $s=0$. The author obtains here a continuation formula near $z=1$ for the series ${}_{p+1}{F}_{p}$ for arbitrary integral $p$ and unrestricted $s$.

##### MSC:
 33C20 Generalized hypergeometric series, ${}_{p}{F}_{q}$ 33C05 Classical hypergeometric functions, ${}_{2}{F}_{1}$