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with its analytic continuation, will define a function which is one-valued over the remainder of the plane of x. The number of roots of this function between 0 and 1 was ascertained first by Stieltjes f and Hilbert J for the special case in which a = — n, when the series reduces to a polynomial. Later the determination for the general case was effected by Klein § in a memoir notable both for its results and for its method. The number of roots between 0 and — oo can be obtained from Klein's results by means of the equation

A determination of the number of real and imaginary roots of the hypergeometric series