Multiple orthogonal polynomials, 𝑑-orthogonal polynomials, production matrices, and branched continued fractions

I analyze an unexpected connection between multiple orthogonal polynomials, d d -orthogonal polynomials, production matrices and branched continued fractions. This work can be viewed as a partial extension of Viennot’s combinatorial theory of orthogonal polynomials to the case where the production matrix is lower-Hessenberg but is not necessarily tridiagonal.