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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
76 Commutation Relations, Normal Ordering, and Stirling Numbers
Note the dependence on x! Thus, we should have better indicated this dependence by
writing S
p,q
(n, k, x) in (3.64). The recurrence relation (3.65) is closely related to (3.63).
In fact, writing
ˆ
S
p,q
(n, k)=Ψ
p,q
(n, k)
nk
p
, one obtains from (3.63) the relation Ψ
p,q
(n +
1,k)=q
k1
Ψ
p,q
(n, k 1) + p
k
[k]
p,q
Ψ
p,q
(n, k). The numbers S
p,q
(n, k) correspond to the
case α =1 =0,r = 0 of the type II generalized (p, q)-Stirling numbers
˜
S
1,p,q
n,k
(α, β, r)
introduced by Remmel and Wachs [928], which will be considered in Chapter 4.
It is possible to define a (p, q)-exponential function by
e
p,q
(x)=
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Publisher Resources

ISBN: 9781466579897