  The main objects of the investigation presented in this paper are
branched-continued-fraction representations of ratios of contiguous
hypergeometric series and type II multiple orthogonal polynomials on the
step-line with respect to linear functionals or measures whose moments are
ratios of products of Pochhammer symbols. This is an interesting case study of
the recently found connection between multiple orthogonal polynomials and
branched continued fractions that gives a clear example of how this connection
leads to considerable advances on both topics. We obtain new results about
generating polynomials of lattice paths and total positivity of matrices and
give new contributions to the general theory of the connection between multiple
orthogonal polynomials and branched continued fractions. We construct new
branched continued fractions for ratios of contiguous hypergeometric series. We
give conditions for positivity of the coefficients of these branched continued
fractions and we show that the ratios of products of Pochhammer symbols are
generating polynomials of lattice paths for a special case of the same branched
continued fractions. We introduce a family of type II multiple orthogonal
polynomials on the step-line associated with those branched continued
fractions. We present a formula as terminating hypergeometric series for these
polynomials, we study their differential properties, and we explicitly find
their recurrence relation coefficients. Finally, we focus the analysis of the
multiple orthogonal polynomials to the cases where the corresponding
branched-continued-fraction coefficients are all positive. In those cases, the
orthogonality conditions can be written using measures on the positive real
line involving Meijer G-functions and we obtain results about the location of
the zeros and the asymptotic behaviour of the polynomials.
