  This paper is about the positive part $U_q^+$ of the $q$-deformed enveloping
algebra $U_q(\widehat{\mathfrak{sl}}_2)$. The literature contains at least
three PBW bases for $U_q^+$, called the Damiani, the Beck, and the alternating
PBW bases. These PBW bases are related via exponential formulas. In this paper,
we introduce an exponential generating function whose argument is a power
series involving the Beck PBW basis and an integer parameter $m$. The cases
$m=2$ and $m=-1$ yield the known exponential formulas for the Damiani and
alternating PBW bases, respectively. The case $m=1$ appears in the author's
previous paper. In the present paper, we give a comprehensive study of the
generating function for an arbitrary integer $m$. We have two main results. The
first main result gives a factorization of the generating function. In the
second main result, we express the coefficients of the generating function in
closed form.
