  We introduce a new family of multiple orthogonal polynomials satisfying
orthogonality conditions with respect to two weights $(w_1,w_2)$ on the
positive real line, with $w_1(x)=x^\alpha e^{-x}$ the gamma density and $w_2(x)
= x^\alpha E_{\nu+1}(x)$ a density related to the exponential integral
$E_{\nu+1}$. We give explicit formulas for the type I functions and type II
polynomials, their Mellin transform, Rodrigues formulas, hypergeometric series
and recurrence relations. We determine the asymptotic distribution of the
(scaled) zeros of the type II multiple orthogonal polynomials and make a
connection to random matrix theory. Finally, we also consider a related family
of mixed type multiple orthogonal polynomials.
