Let $\lambda$ and $\mu$ be partitions of $n$. As I understand it the (transformed) Hall-Littlewood function $H_\mu(t)$ is defined by the expansion
\begin{equation} H_\mu(t) \, := \ \sum_{\lambda \, \vdash n} \, K_{\lambda \mu}(t) \, s_\lambda \end{equation}
where $K_{\lambda \mu}(t)$ is the Kostka-Foulkes polynomial and $s_\lambda$ is the Schur function. $H_\mu(t)$ interpolates between the Schur function $s_\mu = H_\mu(0)$ and the homogeneous symmetric function $h_\mu = H_\mu(1)$.
Question: What if we "invert" the construction and define instead $S_\mu(t)$ implicitly by
\begin{equation} h_\mu \, = \ \sum_{\lambda \, \vdash n} \, K_{\lambda \mu}(t) \, S_\mu(t) \end{equation}
Is this $t$-deformed Schur function $S_\mu(t)$ considered and/or relevant in the theory of symmetric functions?
thanks, ines.