  For all integers $n\geq1$, let \begin{align*}
  W_n(p,q)=\prod_{j=1}^{n}\left\{e^{-p/j}\left(1+\frac{p}{j}+\frac{q}{j^2}\right)\right\}
\end{align*} and \begin{align*} R_n(p,
q)=\prod_{j=1}^{n}\left\{e^{-p/(2j-1)}\left(1+\frac{p}{2j-1}+\frac{q}{(2j-1)^2}\right)\right\},
\end{align*} where $p$, $q$ are complex parameters. The infinite product
$W_{\infty}(p,q)$ includes the Wallis and Wilf formulas, and also the infinite
product definition of Weierstrass for the gamma function, as special cases. In
this paper, we present asymptotic expansions of $W_n(p,q)$ and $R_n(p, q)$ as
$n\to\infty$. In addition, we also establish asymptotic expansions for the
Wallis sequence.
