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

Multiple orthogonal polynomials associated with branched continued fractions for ratios of hypergeometric series