Capelli operators for spherical superharmonics and the Dougall-Ramanujan\n identity

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

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