Determinant, permanent, and MacMahon's master theorem (Q677807)

Motivated by the \(\alpha\)-extension of the determinant and the permanent introduced by \textit{D. Foata} and \textit{D. Zeilberger} [SIAM J. Discrete Math. 1, No. 4, 425-433 (1988; Zbl 0662.05003)] and \textit{D. Vere-Jones} [Linear Algebra Appl. 111, 119-124 (1988; Zbl 0665.15007)], the author defines a matrix function with two parameters. Basic formulas are established, which contain several known relations between determinants and permanents as special cases. Using formal power series expansions for these functions, a common generalization is developed for MacMahon's master theorem, its \(\alpha\)-extension, and their dual versions.