Total Positivity of Almost-Riordan Arrays: Total Positivity of Almost-Riordan Arrays - He, Tian-Xiao; Słowik, Roksana (Graphs and combinatorics, 2025)
In this paper we study the total positivity of almost-Riordan arrays ( d ( t ) | g ( t ) , f ( t ) ) and establish its necessary conditions and sufficient conditions, particularly, for some well-used formal power series d ( t ). We present a semidirect product of an almost-array and use it to transfer a total positivity problem for an almost-Riordan array to the total positivity problem for a quasi-Riordan array. We find the sequence characterization of total positivity of the almost-Riordan arrays. The production matrix J of an almost-Riordan array ( d | g , f ) is presented so that the total positivity of J implies that of both the almost-Riordan array ( d | g , f ) and the Riordan array ( g , f ). We also present a counterexample to illustrate that this sufficient condition is not necessary. If the production matrix J is tridiagonal, then the expressions of its principal minors are given. By using these expressions, we find a sufficient and necessary condition of the total positivity of almost-Riordan arrays with tridiagonal production matrices. Numerous examples are given to demonstrate our results.