  The "Capelli problem" for the symmetric pairs $(\mathfrak{gl}\times
\mathfrak{gl},\mathfrak{gl})$ $(\mathfrak{gl},\mathfrak{o})$, and
$(\mathfrak{gl},\mathfrak{sp})$ is closely related to the theory of Jack
polynomials and shifted Jack polynomials for special values of the parameter.
In this paper, we extend this connection to the Lie superalgebra setting,
namely to the supersymmetric pairs
$(\mathfrak{g},\mathfrak{g}):=(\mathfrak{gl}(m|2n),\mathfrak{osp}(m|2n))$ and
$(\mathfrak{gl}(m|n)\times\mathfrak{gl}(m|n),\mathfrak{gl}(m|n))$, acting on
$W:=S^2(\mathbb C^{m|2n})$ and $\mathbb C^{m|n}\otimes(\mathbb C^{m|n})^*$. We
also give an affirmative answer to the abstract Capelli problem for these
cases.
