A generalization of permanents and determinants
Let \(\alpha\) be an arbitrary real number and let \(A=(a_{ij})\) be a \(k\times k\)-matrix. Then the \(\alpha\)-permanent of A is defined to be the expression \(^{\alpha}| A|^{\alpha}:=\sum_{\sigma \in S_ k}\alpha^{m(\sigma)}a_{1\sigma (1)}\cdot...\cdot a_{k\sigma(k)}\), where \(S_ k\) denotes the symmetric group of order \(k\) and where \(m(\sigma)\) is the number of distinct irreducible cycles of the permutation \(\sigma\). Let \(z_ 1,...,z_ k\) be k independent complex variables and let \(Z=diag(z_ 1,...,z_ k)\). The author proves that \[ (*)\quad [\det (I-ZA)]^{-\alpha}=1+\sum^{k}_{i=1}z_ i(\alpha a_{ii})+ 1/2!\sum^{k}_{i=1} \sum^{k}_{j=1} z_ iz_ j^{\alpha} \left| \begin{matrix} a_{ii} & a_{ij}\\ a_{ji} & a_{jj} \end{matrix} \right|^{\alpha}+...., \] the series converging whenever the modulus of the maximum eigenvalue of ZA is smaller than 1. In the case \(\alpha =1\) one gets the expansion of the reciprocal characteristic polynomial in terms of (sub)permanents, which is the main result due to the same author [ibid. 63, 267-270 (1984; Zbl 0552.15004)]. This also generalizes the so-called MacMahon's Master Theorem, \textit{P. A. MacMahon} [Combinatory Analysis. Vol. I, II (1960; Zbl 0101.25102) and Collected papers. Volume I: Combinatorics. (1978; Zbl 0557.01015)] . At the end of this paper, an interesting application of the formula (*) in probability theory is mentioned.
scientific article; zbMATH DE number 4087840