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

We evaluate the sum of Gauss hypergeometric functions $$S(\mu ,c;x)=\sum _{k\ge 0}\text{bl}(\frac{1-x}{1+\mu }\text{br}{)}^k{}_2{F}_1(\text{fs}k+\text{fs},\text{fs}k+1;c;x)$$ for $x\in [-1,1]$ and positive parameters $\mu$ and $c$. The domain of absolute convergence of this series is established by determining the growth of the hypergeometric function for $k\to +\mathrm{\infty }$. 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.