We introduce the quantum Berezinian for the quantum affine superalgebra $\mathrm{ U}_q(\widehat{ \mathfrak{ gl}}_{ M|N})$ and show that the coefficients of the quantum Berezinian belong to the center of $\mathrm{ U}_q(\widehat{ \gl}_{ M|N})$. We also construct another family of central elements which can be expressed in the quantum Berezinian by a Liouville-type theorem. Moreover, we prove analogues of the Jacobi identities, the Schur complementary theorem, the Sylvester theorem and the MacMahon Master theorem for the generator matrices of $\mathrm{ U}_q(\widehat{ \gl}_{ M|N})$.
Categories: math.QA math.RT