Log-concavity and almost log-convexity of the cycle index polynomials were proved by Bender and Canfield [ J. Combin. Theory Ser. A, 74 (1996), pp. 57--70]. Schirmacher [ J. Combin. Theory Ser. A, 85 (1999), pp. 127--134] extended them to $q$-log-concavity and almost $q$-log-convexity. Motivated by these, we consider the stronger properties total positivity from the Toeplitz matrix and Hankel matrix. By using exponential Riordan array methods, we give some criteria for total positivity of the triangular matrix of coefficients of the generalized cycle index polynomials, the Toeplitz matrix and Hankel matrix of the polynomial sequence in terms of the exponential formula, the logarithmic formula, and the fractional formula, respectively. Finally, we apply our criteria to some triangular arrays satisfying some recurrence relations, including Bessel triangles of two kinds and their generalizations, the Lah triangle and its generalization, the idempotent triangle, and some triangles related to binomial coefficients, rook polynomials, and Laguerre polynomials. We not only get total positivity of these lower-triangles, and $q$-Stieltjes moment properties and 3-$q$-log-convexity of their row-generating functions, but also prove that their triangular convolutions preserve the Stieltjes moment property. In particular, we solve a conjecture of Sokal on the $q$-Stieltjes moment property of rook 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 a 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 criteria 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 $q$-Schröder 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 Schröder triangle, and so on, which extend many known results for $q$-log-convexity.