Coefficientwise Hankel-total positivity of row-generating polynomials for the m-Jacobi-Rogers triangle [article]

Bao-Xuan Zhu
2022 arXiv   pre-print
The aim of this paper is to study the criteria for the row-generating polynomial sequence of the m-Jacobi-Rogers triangle being coefficientwise Hankel-totally positive and their applications. Using the theory of production matrices, we gain a criterion for the coefficientwise Hankel-total positivity of the row-generating polynomial sequence of the m-Jacobi-Rogers triangle. This immediately implies that the corresponding m-Jacobi-Rogers triangular convolution preserves the Stieltjes moment
more » ... ty 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 coefficientwise Hankel-total positivity for the Catalan-Stieltjes matrices. 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 for exponential Rirodan arrays. In addition, we also derive some results for the coefficientwise Hankel-total positivity in terms of compositional functions and m-branched Stieltjes continued fractions. We apply our results to many combinatorial polynomials in a unified manner. In particular, we also solve some conjcetures proposed by Sokal.
arXiv:2202.03793v1 fatcat:vlcl67ceyrd4jpswirdc23qo6m