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The aim of this paper is to study the criteria for the row-generatingpolynomial sequence of the $m$-Jacobi-Rogers triangle being coefficientwiseHankel-totally positive and their applications.  Using the theory of production matrices, we gain a criterion for thecoefficientwise Hankel-total positivity of the row-generating polynomialsequence of the $m$-Jacobi-Rogers triangle. This immediately implies that thecorresponding $m$-Jacobi-Rogers triangular convolution preserves the Stieltjesmoment property of sequences and its zeroth column sequence is coefficientwiseHankel-totally positive and log-convex of higher order in all theindeterminates. In consequence, for $m=1$, we immediately obtain some resultson coefficientwise Hankel-total positivity for the Catalan-Stieltjes matrices.For the general $m$, combining our criterion and a function satisfying anautonomous differential equation, we present different criteria forcoefficientwise Hankel-total positivity of the row-generating polynomialsequence for exponential Rirodan arrays. In addition, we also derive someresults for the coefficientwise Hankel-total positivity in terms ofcompositional functions and $m$-branched Stieltjes continued fractions. Weapply our results to many combinatorial polynomials in a unified manner. Inparticular, we also solve some conjcetures proposed by Sokal.

Coefficientwise Hankel-total positivity of row-generating polynomials for the $m$-Jacobi-Rogers triangle