
A Generalization of Touchard Polynomials 395
Let us turn to the case m = −1. Recall that the q-deformed Bessel polynomials were
discussed in Section 9.4.3.
Proposition 10.49 The q-deformed Touchard polynomials of order m = −1 can be ex-
pressed by q-deformed Bessel polynomials as
T
(−1)
n|q
(x)=x
−n
q
−(n−1)
2
y
n−1
−
1
x
; q
. (10.45)
Proof Theorem 10.45 implies that T
(−1)
n|q
(x)=x
−2n
B
2;−q
−1
|n|q
−1 (x),whereweused
that [−1]
q
= −q
−1
. The same argument as in Lemma 8.98 shows that B
s;h|n|q
(x)=
h
n
B
s;1|n|q
(
x
h
). Thus, T
(−1)
n|q
(x)=x
−2n
q
−n
B
2;−1|n|q
−1
(qx). Inserting the expression resulting
for B
2;−1|n|q
−1
(qx) from (9.61) gives, after some simplifications, the assertion.