
Generalizations of Stirling Numbers 129
Definition 4.116 (Xu) Let ¯α =(α
i
)
i≥0
and
¯
β =(β
i
)
i≥0
be two sequences of complex
numbers. The generalized Stirling numbers of the first and of the second kind are defined
by
(t|
¯
β)
n
=
n
k=0
s(n, k;¯α,
¯
β,r)(t + r|¯α)
k
,
(t|¯α)
n
=
n
k=0
S(n, k;¯α,
¯
β,r)(t − r|
¯
β)
k
.
Comparing this definition with Definition 4.49 for the Stirling-type pairs of Hsu and Shiue,
we see that if α
k
= α and β
k
= β for all k, the above numbers reduce to the generalized
Stirling numbers of Hsu and Shiue, that is,
s(n, k;¯α,
¯
β,r)=S
2
(n, k; α, β, r)andS(n, k;¯α,
¯
β,r)=S
1
(n, k; α, β, r).
From the definition, one has S(n, k;¯α,
¯
β,r)=s(n, k;
¯
β, ¯α, −r), so it suffices ...