Hausdorff moment sequences and hypergeometric functions
Abstract
P\'olya in 1926 showed that the hypergeometric function F(z)=2F1(a,b;c;z) has a totally monotone sequence as its coefficients; that is, F is the generating function of a Hausdorff moment sequence, when 0 a 1 and 0 b c. In this paper, we give a complete characterization of such hypergeometric functions F in terms of complex parameters a,b,c. To this end, we study the class of general properties of generating functions of Hausdorff moment sequences and, in particular, we provide a sufficient condition for the class by making use of a Phragm\`en-Lindel\"of type theorem. As an application, we give also a necessary and sufficient condition for a shifted hypergeometric function to be universally starlike.
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