$c_\lambda$ given in Definition \[dfcl\] that are characterized by suitable symmetry and vanishing conditions (see Theorem \[thm-unqclam\]). Furthermore, in Theorem \[MAINTHM\] we show that the top degree homogeneous term of the eigenvalue polynomial $c_\lambda$ is equal to the spherical polynomial $d_\lambda$ given in Definition \[dfbr\]. This extends the result proved by Knop and Sahi in [@KnopSahi] explained above. The corresponding result for the superpair $(\gl(m|n)\times \gl(m|n),\gl(m|n))$ is Theorem \[THMAppB2\]. In this case the spherical and eigenvalue polynomials turn out to be the well known *supersymmetric Schur polynomials* and *shifted supersymmetric Schur* polynomials [@Molev].

Our third main goal is to establish a precise relation between our eigenvalue polynomials $c_\lambda$ and certain polynomials called *shifted super Jack polynomials*. In connection with the eigenstates of the deformed Calogero–Moser–Sutherland operators [@SerVes], Sergeev and Veselov define a family of $(m+n)$-variable polynomials $SP^*_\flat$, parametrized by $(m,n)$-hook partitions $\flat$ (see Definition \[dfnhookprn\]). In the case of the superpair $(\gl(m|2n),\g{osp}(m|2n))$, in Proposition \[prpQlam\] and Theorem \[thmconnSV\], we prove that the $c_\lambda$ and the $SP^*_\flat$ are related by the *Frobenius transform*, see Definition \[FrobT\]. The corresponding statement for the pair $(\gl(m|n)\times\gl(m|n),\gl(m|n))$ is Theorem \[THMAB3\].

It is worth mentioning that in [@SerVes], the shifted super Jack polynomials are obtained as 