  We focus on the generating series for the rational special values of
Pellarin's $L$-series in $1 \leq s \leq 2(q-1)$ indeterminates, and using
interpolation polynomials we prove a closed form formula relating this
generating series to the Carlitz exponential, the Anderson-Thakur function, and
the Anderson generating functions for the Carlitz module. We draw several
corollaries, including explicit formulae and recursive relations for Pellarin's
$L$-series in the same range of $s$, and divisibility results on the numerators
of the Bernoulli-Carlitz numbers by monic irreducibles of degrees one and two.
