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

The main objects of the investigation presented in this paper are branched-continued-fraction representations ratios contiguous hypergeometric series and type II multiple orthogonal polynomials on step-line with respect to linear functionals or measures whose moments products Pochhammer symbols. This is an interesting case study recently found connection between branched continued fractions that gives a clear example how leads considerable advances both topics. We start by obtaining new results about generating lattice paths total positivity matrices giving contributions general theory emphasis its application analysis polynomials. Then, we construct for series. give conditions coefficients these show symbols special under study. Next, introduce family associated those fractions. present formula as terminating polynomials, their differential properties, find explicit recurrence relation satisfied them. Finally, focus cases where corresponding all positive. In cases, orthogonality can be written using positive real line involving Meijer G-functions obtain location zeros asymptotic behaviour Specialisations studied here include classical Laguerre, Jacobi, Bessel Nikishin systems two modified functions, confluent Gauss' function, used investigate singular values Ginibre random well r-orthogonal polynomial sequence constant any integer r particular instances Jacobi-Piñeiro