  The frame set of a function $g\in L^2(\mathbb{R})$ is the subset of all
parameters $(a, b)\in \mathbb{R}^2_+$ for which the time-frequency shifts of
$g$ along $a\mathbb{Z}\times b\mathbb{Z}$ form a Gabor frame for
$L^2(\mathbb{R}).$ In this paper, we investigate the frame set of a class of
functions that we call \emph{generalized $B-$splines} and which includes the
$B-$splines. In particular, we add many new points to the frame sets of these
functions. In the process, we generalize and unify some recent results on the
frame sets for this class of functions.
