An Alternative Generating Function for $k$-Regular Partitions

We construct a $k$-fold $q$-series as a generating function of $k$-regular partitions for each positive integer $k$. The $k=1$ case is one of Euler’s $q$-series identities pertaining to the partitions into distinct parts. The construction is combinatorial. Although we find a connection to Bessel polynomials in the $k=2$ case, this note is certainly not a study of Bessel polynomials and their $q$-analogs.