In his proof of the irrationality of $\zeta(2)$ and $\zeta(3)$, Ap\'ery
introduced the numbers $A_n$ and $B_n$ in the form of binomial sums. We show
that the two sequences $\{A_n\}_{n=0}^\infty$ and $\{B_n\}_{n=0}^\infty$ are
log-convex in the senses that $A_n^2< A_{n-1}A_{n+1}$ and $B_n^2<
B_{n-1}B_{n+1}$ for $n\geq 1$. Recently, Zudilin defined the numbers $U_n$ as a
double sum of products of binomial coefficients in connection with $\zeta(4)$.
We show that the sequence $\{U_n\}_{n=0}^\infty$ of Zudilin numbers is also
log-convex.

We present an operator approach to Rogers-type formulas and Mehler's formulas
for the Al-Salam-Carlitz polynomials $U_n(x,y,a;q)$. By using the q-exponential
operator, we obtain a Rogers-type formula which leads to a linearization
formula. With the aid of a bivariate augmentation operator, we get a simple
derivation of Mehler's formula due to by Al-Salam and Carlitz, which requires a
terminating condition on a ${}_3\phi_2$ series. By means of the Cauchy
companion augmentation operator, we obtain Mehler's formula in a similar form,
but it does not need the terminating condition.