  An identity involving basic Bessel functions and Al-Salam--Chihara
polynomials is proved for which we recover Graf's addition formula for the
Bessel function as the base $q$ tends to $1$. The corresponding product formula
is derived. Some known identities for Jackson's $q$-Bessel functions are
obtained as limiting cases. As special cases we prove identities for
$q$-Charlier polynomials.
