2 research outputs found
Generalized -Stirling numbers and normal ordering
The normal ordering coefficients of strings consisting of which satisfy
() are considered. These coefficients are studied
in two contexts: first, as a multiple of a sequence satisfying a generalized
recurrence, and second, as -analogues of rook numbers under the row creation
rule introduced by Goldman and Haglund. A number of properties are derived,
including recurrences, expressions involving other -analogues and explicit
formulas. We also give a Dobinsky-type formula for the associated Bell numbers
and the corresponding extension of Spivey's Bell number formula. The
coefficients, viewed as rook numbers, are extended to the case
via a modified rook model.Comment: New section on q-Bell numbers added, extended to case $s\in\mathbb R
Rook theoretic proofs of some identities related to Spivey's Bell number formula
We use rook placements to prove Spivey's Bell number formula and other
identities related to it, in particular, some convolution identities involving
Stirling numbers and relations involving Bell numbers. To cover as many special
cases as possible, we work on the generalized Stirling numbers that arise from
the rook model of Goldman and Haglund. An alternative combinatorial
interpretation for the Type II generalized -Stirling numbers of Remmel and
Wachs is also introduced in which the method used to obtain the earlier
identities can be adapted easily.Comment: 9 page