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