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We define an infinite sequence of generalizations, parametrized by an integer m > 1, of the Stieltjes–Rogers and Thron–Rogers polynomials; they arise as the power-series expansions of some branched continued fractions, and as the generating polynomials for m-Dyck and m-Schröder paths with height-dependent weights. We prove that all of these sequences of polynomials are coefficientwise Hankel-totally positive, jointly in all the (infinitely many) indeterminates. We then apply this theory toarXiv:1807.03271v2 fatcat:mhzpgcz5fzawxk5uzp72imemse