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We establish a duality of the non-periodic Gaudin model associated withsuperalgebra $\mathfrak{gl}_{m|n}$ and the non-periodic Gaudin model associatedwith algebra $\mathfrak{gl}_k$.  The Hamiltonians of the Gaudin models are given by expansions of a Berezinianof an $(m+n)\times(m+n)$ matrix in the case of $\mathfrak{gl}_{m|n}$ and of acolumn determinant of a $k\times k$ matrix in the case of $\mathfrak{gl}_k$. Weobtain our results by proving Capelli type identities for both cases andcomparing the results.

The duality of $\mathfrak{gl}_{m|n}$ and $\mathfrak{gl}_k$ Gaudin models