algorithms. Section 9.4 extends those results to the determinants #_ =
IA
I P(A) -WI PiA -' I whose a -> 0' values are required for use in Propositions 7.5-6.
The utility of these representations depends upon the ability to extract useful relationships
between the C'(m,r)'s from the general form represented by Eq. 9.27 and Eq. 9.28. The
following paragraphs examine the combinatorial determinants in the general forms pro-
vided by Eq. 9.27 and Eq. 9.28 and deduce some of the key relationships. The purpose of
this effort is to provide a foundation for extending the stationary distribution
representation methodology developed in Sections 9.2-9.4.
First, if the enabling condition for Eq. 9.22 is satisfied (i.e. M < N), then every ele-
ment in the combinatorial determinant of Eq. 9.27 is either zero or it is the combinatorial
determinant corresponding to the order zero coefficient for some state in S'. Thus, every
coefficient of the form represented by Eq. 9.27 can be written as sums and products of
order zero coefficients. An analogous conclusion applies to Eq. 9.28.
Second, it is clear from Eq. 9.27 that nonzero order differentiation of any two or
more rows of -I P -II =I P-II -I P i| in an identical pattern (e.g. r' : 0 rj, = rj for
n, # n2) introduces identical rows into the combinatorial determinant, and thus makes
C'(m,r') 0. Consequently, no monomial terms corresponding to any r' with identical
nonzero rows survive in the expansion of P,-1I = P-II -| P-1I. An identical con-
clusion applies to the coefficients of- P -, = P(mA)' -'l -i' of which Eq.
9.28 is an exemplar.
A very important class of coefficient identities derives from transpositions of non-
zero rows and columns of the differentiation order array. The resulting identities are very
closely connected to the algebra of symmetric and alternating polynomials, and to an
associated determinant concept called alternants, of which Vandermonde determinants
are a special case. Appendix C is provided to support the following paragraphs.
From the form of the combinatorial determinants in Eq. 9.27 and Eq. 9.28, it is clear
that exchanging any two of the k rows indexed by row indices n e K is equivalent to