The Real Powers of the Convolution of a Gamma Distribution and a Bernoulli Distribution

In this paper, we essentially compute the set of $x,y>0$ such that the mapping z \longmapsto \Big{(}1-r+r e^z\Big{)}^x \Big{(}\dis\frac{\lambda}{\lambda-z}\Big{)}^{y} is a Laplace transform. If $X$ and $Y$ are two independent random variables which have respectively Bernoulli and Gamma distributions, we denote by $\mu$ the distribution of $X+Y.$ The above problem is equivalent to finding the set of $x>0$ such that $\mu^{{\ast}x}$ exists.