\lambda +\rho \right) $, where $\varphi _{\mu }$ is a symmetric polynomial and $\rho $ is a certain “rho-shift". Although [@KostantSahi1], [@KostantSahi2] are written in the context of symmetric spaces, similar ideas work for the multiplicity-free setting, see [@Knop].

In [@Sahi], it is shown that the polynomials $\varphi _{\mu }$ are uniquely characterized by certain vanishing conditions. In fact [@Sahi] considers a general class of polynomials, depending on several parameters, and in [@KnopSahi] it is shown that a one-parameter subfamily of these polynomials is closely related to *Jack polynomials*. More precisely, the Knop–Sahi polynomials are inhomogeneous polynomials, but their top degree terms are the (homogenenous) Jack polynomials. In the special case where the value of the parameter $\theta$ corresponds to a symmetric space, the Knop–Sahi polynomials are the eigenvalue polynomials $\varphi _{\lambda }$, and the Jack polynomials are the spherical polynomials in $V_{\lambda }$. Many properties of the Knop–Sahi polynomials were subsequently proved by Okounkov and Olshanski [@Okounkov], who worked with a slight modification of these polynomials which they call *shifted Jack polynomials*.

The goal of our paper is to extend this circle of ideas to the Lie superalgebra setting. The Jack polynomials for $\theta=1,\frac{1}{2},2$ are spherical polynomials of the 