Positivity of iterated sequences of polynomials

In this paper, we present some criteria for the $2$-$q$-log-convexity and $3$-$q$-log-convexity of combinatorial sequences, which can be regarded as the first column of certain infinite triangular array $[A_{n,k}(q)]_{n,k\geq0}$ of polynomials in $q$ with nonnegative coefficients satisfying the recurrence relation $$A_{n,k}(q)=A_{n-1,k-1}(q)+g_k(q)A_{n-1,k}(q)+h_{k+1}(q)A_{n-1,k+1}(q).$$ Those criterions can also be presented by continued fractions and generating functions. These allow a unified treatment of the $2$-$q$-log-convexity of alternating Eulerian polynomials, $2$-log-convexity of Euler numbers, and $3$-$q$-log-convexity of many classical polynomials, including the Bell polynomials, the Eulerian polynomials of Types $A$ and $B$, the $q$-Schr\"{o}der numbers, $q$-central Delannoy numbers, the Narayana polynomials of Types $A$ and $B$, the generating functions of rows in the Catalan triangles of Aigner and Shapiro, the generating functions of rows in the large Schr\"oder triangle, and so on, which extend many known results for $q$-log-convexity.