  Ordering identities in the Weyl-Heisenberg algebra generated by single-mode
boson operators are investigated. A boson string composed of creation and
annihilation operators can be expanded as a linear combination of other such
strings, the simplest example being a normal ordering. The case when each
string contains only one annihilation operator is already combinatorially
nontrivial. Two kinds of expansion are derived: (i) that of a power of a string
$\Omega$ in lower powers of another string $\Omega'$, and (ii) that of a power
of $\Omega$ in twisted versions of the same power of $\Omega'$. The expansion
coefficients are shown to be, respectively, generalized Stirling numbers of Hsu
and Shiue, and certain generalized Eulerian numbers. Many examples are given.
These combinatorial numbers are binomial transforms of each other, and their
theory is developed, emphasizing schemes for computing them: summation
formulas, Graham-Knuth-Patashnik (GKP) triangular recurrences, terminating
hypergeometric series, and closed-form expressions. The results on the first
type of expansion subsume a number of previous results on the normal ordering
of boson strings.
