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An analytic continuation of the hypergeometric series. (English) Zbl 0614.33005
The points $z=\left(1±i\sqrt{3}\right)$ of the complex z-plane are on the boundary for each of the convergence domains of the various hypergeometric series which appear in the transformation or continuation formulas of the hypergeometric function ${}_{2}{F}_{1}\left(a,b;c;z\right)$. This paper presents a continuation formula containing series in powers of 1/(z-$\right)$ with the convergence domain $|z-|>$, which contains the two points in question in its interior. The coefficients of the power series are determined by a three-term recurrence relation and are represented explicitly in terms of terminating hypergeometric series. If $2c=a+b+1$, then one term of recurrence relation disappears and the series become hypergeometric series.

##### MSC:
 33C05 Classical hypergeometric functions, ${}_{2}{F}_{1}$ 34M99 Differential equations in the complex domain 30B40 Analytic continuation (one complex variable) 34A30 Linear ODE and systems, general
##### Keywords:
hypergeometric series