  Total positivity of matrices is deeply studied and plays an important role in
various branches of mathematics. The aim of this paper is to study the criteria
for coefficientwise Hankel-total positivity of the row-generating polynomials
of generalized $m$-Jacobi-Rogers triangles and their applications.
  Using the theory of production matrices, we present the criteria for
coefficientwise Hankel-total positivity of the row-generating polynomials of
the output matrices of certain production matrices. In particular, we gain a
criterion for coefficientwise Hankel-total positivity of the row-generating
polynomial sequence of the generalized $m$-Jacobi-Rogers triangle. This
immediately implies that the corresponding generalized $m$-Jacobi-Rogers
triangular convolution preserves the Stieltjes moment property of sequences and
its zeroth column sequence is coefficientwise Hankel-totally positive and
log-convex of higher order in all the indeterminates. In consequence, for
$m=1$, we immediately obtain some results on Hankel-total positivity for the
Catalan-Stieltjes matrices. In particular, we in a unified manner apply our
results to some combinatorial triangles or polynomials including the
generalized Jacobi Stirling triangle, a generalized elliptic polynomial, a
refined Stirling cycle polynomial and a refined Eulerian polynomial. For the
general $m$, combining our criterion and a function satisfying an autonomous
differential equation, we present different criteria for coefficientwise
Hankel-total positivity of the row-generating polynomial sequence of
exponential Rirodan arrays. In addition, we also derive some results for
coefficientwise Hankel-total positivity in terms of compositional functions and
$m$-branched Stieltjes-type continued fractions. We apply our results to many
combinatorial polynomials and solve some conjcetures proposed by Sokal.
