Total Positivity of Almost-Riordan Arrays

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.