2024/11/07 by Sugawa, Toshiyuki, Wang, Li-Mei
#Complex Variables (math.CV) #FOS: Mathematics #Primary 33C05 #Secondary 30E05
paper · doi:10.48550/arxiv.2411.04345
Pólya in 1926 showed that the hypergeometric function F(z)=\null2F1(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 Phragmèn-Lindelöf type theorem. As an application, we give also a necessary and sufficient condition for a shifted hypergeometric function to be universally starlike.