Boson Operator Ordering Identities from Generalized Stirling and Eulerian Numbers

Ordering identities in the Weyl-Heisenberg algebra generated by single-modeboson operators are investigated. A boson string composed of creation andannihilation operators can be expanded as a linear combination of other suchstrings, the simplest example being a normal ordering. The case when eachstring contains only one annihilation operator is already combinatoriallynontrivial. 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 powerof $\Omega$ in twisted versions of the same power of $\Omega'$. The expansioncoefficients are shown to be, respectively, generalized Stirling numbers of Hsuand Shiue, and certain generalized Eulerian numbers. Many examples are given.These combinatorial numbers are binomial transforms of each other, and theirtheory is developed, emphasizing schemes for computing them: summationformulas, Graham-Knuth-Patashnik (GKP) triangular recurrences, terminatinghypergeometric series, and closed-form expressions. The results on the firsttype of expansion subsume a number of previous results on the normal orderingof boson strings.