In this paper, we extend the results of Grantcharov and Robitaille in 2021 on mixed tensor products and Capelli determinants to the superalgebra setting. Specifically, we construct a family of superalgebra homomorphisms φR : U(gl(m+1|n)) → D'(m|n) ⊗ U(gl(m|n)) for a certain space of differential operators D'(m|n) indexed by a central element R of D'(m|n) ⊗ U(gl(m|n)). We then use this homomorphism to determine the image of Gelfand generators of the center of U(gl(m+1|n)). We achieve this by first relating φR to the corresponding Harish-Chandra homomorphisms and then proving a super-analog of Newton's formula for gl(m) relating Capelli generators and Gelfand generators. We also use the homomorphism φR to obtain representations of U(gl(m+1|n)) from those of U(gl(m|n)), and find conditions under which these inflations are simple. Finally, we show that for a distinguished central element R₁ in D'(m|n)⊗ U(gl(m|n)), the kernel of φR₁ is the ideal of U(gl(m+1|n)) generated by the first Gelfand invariant G₁.
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Erat et al. (2024) studied this question.