This exact version of a "generalized determinantpermanent" is called $\beta$-extension in this Foata-Zeilberger paper (see also my paper for the algebraic context and further non-commutative generalizations, Cartier-Foata style).
This exact version of a "generalized determinant" is called $\beta$-extension in this Foata-Zeilberger paper (see also my paper for the algebraic context and further non-commutative generalizations, Cartier-Foata style).