A note on the quantum Berezinian for the double Yangian of the Lie superalgebra ${\mathfrak{g}\mathfrak{l}}_{m\mathrm{\mid }n}$

In this note, we generalize the notion of quantum Berezinian to the double Yangian $\mathrm{D}\mathrm{Y}({\mathfrak{g}\mathfrak{l}}_{m\mathrm{\mid }n})$ of the Lie superalgebra ${\mathfrak{g}\mathfrak{l}}_{m\mathrm{\mid }n}$. We show that its coefficients form a family of algebraically independent topological generators of the center of $\mathrm{D}\mathrm{Y}({\mathfrak{g}\mathfrak{l}}_{m\mathrm{\mid }n})$.