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 ...