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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
A Generalization of the Weyl Algebra 327
see also (8.93). Since {S
1;1
, S
2;1
} is a dual pair for which one has orthogonality relations
(see Proposition 8.120), the same is true for Bessel numbers, that is, one has that
m
k=n
B(m, k)b(k, n)=
m
k=n
b(m, k)B(k, n)=δ
m,n
.
Of course, these relations are well-known [528, 1170] (for example, in [1170] they were de-
rived via exponential Riordan arrays and Lagrange inversion). Let us summarize the above
observations in the following theorem.
Theorem 8.124 The dual pair {S
1;1
(n, k), S
2;1
(n, k)} is given by the arrays of Bessel
numbers of the second and the first kind {B(n, k),b(n, k)}, that is, S
1;1
(n, k)=B(n, k)
and S
2;1
(n, k)=b(n, k).
Now, we discuss the above results in connection with normal ordering of ...
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Publisher Resources

ISBN: 9781466579897