  We investigate several families of multiple orthogonal polynomials associated
with weights for which the moment generating functions are hypergeometric
series with slightly varying parameters. The weights are supported on the unit
interval, the positive real line, or the unit circle and the multiple
orthogonal polynomials are generalizations of the Jacobi, Laguerre or Bessel
orthogonal polynomials. We give explicit formulas for the type I and type II
multiple orthogonal polynomials and study some of their properties. In
particular, we describe the asymptotic distribution of the (scaled) zeros of
the type II multiple orthogonal polynomials via the free convolution. Essential
to our overall approach is the use of the Mellin transform. Finally, we discuss
two applications. First, we show that the multiple orthogonal polynomials
appear naturally in the study of the squared singular values of (mixed)
products of truncated unitary random matrices and Ginibre matrices. Secondly,
we use the multiple orthogonal polynomials to simultaneously approximate
certain hypergeometric series and to provide an explicit proof of their
$\mathbb{Q}$-linear independence.
