  Recently S. Gerhold and R. Garra-F. Polito independently introduced a new
function related to the special functions of Mittag-Leffler family. This
function is a generalization of the function studied by E. Le Roy in the period
1895-1905 in connection with the problem of analytic continuation of power
series with a finite radius of convergence. In our note we obtain two integral
representations of this special function, calculate its Laplace transform,
determine an asymptotic expansion of this function on the negative semi-axis
(in the case of an integer third parameter $\gamma$) and provide its
continuation to the case of a negative first parameter $\alpha$. An asymptotic
result is illustrated by numerical calculations. Discussion on possible further
studies and open questions are also presented.
