  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.
