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

## TL;DR

This paper evaluates a weighted sum of Gauss hypergeometric functions, establishes its convergence domain, and connects it to Galton-Watson processes, introducing a new class of distributions with power tails.

## Contribution

It provides a closed-form evaluation of a hypergeometric sum and links it to stochastic processes, introducing novel distributions with power-law tails.

## Key findings

- Derived the convergence domain of the hypergeometric sum.
- Connected the sum to Galton-Watson branching processes.
- Introduced a new class of positive integer-valued distributions.

## Abstract

We evaluate the sum of Gauss hypergeometric functions \[S(\mu,c;x)=\sum_{k\geq 0} \bl(\frac{1-x}{1+\mu}\br)^k\,{}_2F_1(\fs k+\fs, \fs k+1;c;x)\] 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 $k\to+\infty$. 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.