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Commutation Relations, Normal Ordering, and Stirling Numbers
book

Commutation Relations, Normal Ordering, and Stirling Numbers

by Toufik Mansour, Matthias Schork
September 2015
Intermediate to advanced content levelIntermediate to advanced
528 pages
19h 34m
English
Chapman and Hall/CRC
Content preview from Commutation Relations, Normal Ordering, and Stirling Numbers
Generalizations of Stirling Numbers 129
Definition 4.116 (Xu) Let ¯α =(α
i
)
i0
and
¯
β =(β
i
)
i0
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, kα,
¯
β,r)(t + r|¯α)
k
,
(t|¯α)
n
=
n
k=0
S(n, kα,
¯
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, kα,
¯
β,r)=S
2
(n, k; α, β, r)andS(n, kα,
¯
β,r)=S
1
(n, k; α, β, r).
From the definition, one has S(n, kα,
¯
β,r)=s(n, k;
¯
β, ¯α, r), so it suffices ...
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Publisher Resources

ISBN: 9781466579897