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An analytic continuation of the hypergeometric series. (English) Zbl 0614.33005
The points $z=(1\pm i\sqrt{3})$ 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 ${}\sb 2F\sb 1(a,b;c;z)$. This paper presents a continuation formula containing series in powers of 1/(z-$)$ with the convergence domain $\vert z-\vert >$, 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.

33C05Classical hypergeometric functions, ${}_2F_1$
34M99Differential equations in the complex domain
30B40Analytic continuation (one complex variable)
34A30Linear ODE and systems, general
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