Dominating numbers at singular cardinals

We study the generalized dominating number ${\mathfrak{d}}_{\mu }$ at a singular cardinal $\mu$ of cofinality $\kappa$. We show two lower bounds: in ZFC, $\mathrm{c}\mathrm{f}([\mu {]}^{\kappa },\subseteq )\le {\mathfrak{d}}_{\mu }$, and under mild cardinal-arithmetic assumptions, $2^{<\mu }\le {\mathfrak{d}}_{\mu }$. We also clarify when ${\mathfrak{d}}_{\mu }$ can differ from $2^{\mu }$: assuming GCH and $\kappa =\mathrm{c}\mathrm{f}(\mu )>\omega$, a finite-support iteration of Cohen forcing of length ${\mu }^{++}$ yields ${\mathfrak{d}}_{\mu }<2^{\mu }$. On the other hand, for $\kappa =\mathrm{c}\mathrm{f}(\mu )=\omega$, natural $\mu$-cc posets force ${\mathfrak{d}}_{\mu }=2^{\mu }$.