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Total positivity of matrices is deeply studied and plays an important role invarious branches of mathematics. The aim of this paper is to study the criteriafor coefficientwise Hankel-total positivity of the row-generating polynomialsof generalized $m$-Jacobi-Rogers triangles and their applications.  Using the theory of production matrices, we present the criteria forcoefficientwise Hankel-total positivity of the row-generating polynomials ofthe output matrices of certain production matrices. In particular, we gain acriterion for coefficientwise Hankel-total positivity of the row-generatingpolynomial sequence of the generalized $m$-Jacobi-Rogers triangle. Thisimmediately implies that the corresponding generalized $m$-Jacobi-Rogerstriangular convolution preserves the Stieltjes moment property of sequences andits zeroth column sequence is coefficientwise Hankel-totally positive andlog-convex of higher order in all the indeterminates. In consequence, for$m=1$, we immediately obtain some results on Hankel-total positivity for theCatalan-Stieltjes matrices. In particular, we in a unified manner apply ourresults to some combinatorial triangles or polynomials including thegeneralized Jacobi Stirling triangle, a generalized elliptic polynomial, arefined Stirling cycle polynomial and a refined Eulerian polynomial. For thegeneral $m$, combining our criterion and a function satisfying an autonomousdifferential equation, we present different criteria for coefficientwiseHankel-total positivity of the row-generating polynomial sequence ofexponential Rirodan arrays. In addition, we also derive some results forcoefficientwise Hankel-total positivity in terms of compositional functions and$m$-branched Stieltjes-type continued fractions. We apply our results to manycombinatorial polynomials and solve some conjcetures proposed by Sokal.

Coefficientwise Hankel-total positivity of the row-generating polynomials for the output matrices of certain production matrices