  Two well-known $q$-Hermite polynomials are the continuous and discrete
$q$-Hermite polynomials. In this paper we consider a new family of $q$-Hermite
polynomials and prove several curious properties about these polynomials. One
striking property is the connection with $q$-Fibonacci and $q$-Lucas
polynomials. The latter relation yields a generalization of the
Touchard-Riordan formula.
