Capelli operators for spherical superharmonics and the Dougall-Ramanujan identity

Let $(V,omega)$ be an orthosympectic $mathbb Z_2$-graded vector space and let $mathfrak g:=mathfrak{gosp}(V,omega)$ denote the Lie superalgebra of similitudes of $(V,omega)$. When the space $mathscr P(V)$ of superpolynomials on $V$ is emph{not} a completely reducible $mathfrak g$-module, we construct a natural basis $D_lambda$ of Capelli operators for the algebra of $mathfrak g$-invariant superpolynomial superdifferential operators on $V$, where the index set $mathcal P$ is the set of integer partitions of length at most two. We compute the action of the operators $D_lambda$ on maximal indecomposable components of $mathscr P(V)$ explicitly, in terms of Knop-Sahi interpolation polynomials. Our results show that, unlike the cases where $mathscr P(V)$ is completely reducible, the eigenvalues of a subfamily of the $D_lambda$ are emph{not} given by specializing the Knop-Sahi polynomials. Rather, the formulas for these eigenvalues involve suitably regularized forms of these polynomials. In addition, we demonstrate a close relationship between our eigenvalue formulas for this subfamily of Capelli operators and the Dougall-Ramanujan hypergeometric identity. We also transcend our results on the eigenvalues of Capelli operators to the Deligne category $mathsf{Rep}(O_t)$. More precisely, we define categorical Capelli operators ${mathbf D_{t,lambda}}_{lambdainmathcal P}^{}$ that induce morphisms of indecomposable components of symmetric powers of $mathsf V_t$, where $mathsf V_t$ is the generating object of $mathsf{Rep}(O_t)$. We obtain formulas for the eigenvalue polynomials associated to the $left{mathbf D_{t,lambda}right}_{lambdainmathcal P}$ that are analogous to our results for the operators ${D_lambda}_{lambdainmathcal P}^{}$.