The evaluation of a weighted sum of Gauss hypergeometric functions and its connection with Galton-Watson processes

2019/04/30 by Paris, R B, Vinogradov, Vladimir V
#33C05 #34E05 #41A60 #60E05 #60E10 #60J80 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.1904.13176

Abstract

We evaluate the sum of Gauss hypergeometric functions S(μ,c;x)=∑_{k≥ 0} \bl((1-x)/(1+μ)\br)^k ₂F₁(\fs k+\fs, \fs k+1;c;x) for x∈ [-1,1] and positive parameters μ and c. The domain of absolute convergence of this series is established by determining the growth of the hypergeometric function for k→+∞. An application to Galton-Watson branching processes arising in the theory of stochastic processes is presented. A new class of positive integer-valued distributions with power tails is introduced.

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