The real powers of the convolution of a negative binomial distribution and a Bernoulli distribution

For $c>0,$ this note computes essentially the set of $(x,y)$ in $[0,+\infty)^2$ such that the entire series in $z$ defined by $(1+z/c)^x(1-z)^{-y}$ has all its coefficients non-negative. If $X$ and $Y$ are independent random variables which have respectively Bernoulli and negative binomial distributions, denote by $\mu$ the distribution of $X+Y$. The above problem is equivalent to finding the set of $p>0$ such that $\mu^{*p}$ exists; this set is a finite union of intervals and may be the first example of this type in the literature. This gives the final touch to the classification of the natural exponential families with variance functions of Babel type, i.e. of the form $aR(m)+(bm+c)\sqrt{R(m)}$, where $R$ is a polynomial with degree $\leq 2.$