
114 Commutation Relations, Normal Ordering, and Stirling Numbers
4.3.1 The q-Deformation due to Corcino, Hsu, and Tan
Let us start with the q-analog of Stirling numbers which was introduced by Corcino,
Hsu, and Tan [295] and was considered further by Corcino and Barrientos [288]. To motivate
the definition, we follow [295] and consider the factorial t(t−[α])(t −[2α]) ···(t−[(n −1)α])
as an analog of (t|α)
n
, where we denote [α] ≡ [α]
q
=(q
α
−1)/(q −1). Multiplying this with
(q −1)
n
,wegettheproduct(x −q
0
)(x −q
α
) ···(x −q
(n−1)α
), where x = t(q −1) + 1. This
suggests the definition [[t|q
α
]]
n
=
n−1
j=0
(t −q
jα
)forn ≥ 1. For simplicity, we denote a = q
α
with a