
28 Commutation Relations, Normal Ordering, and Stirling Numbers
where the polynomial Δ(x) is called the characteristic polynomial of the recurrence relation
(2.2). The fundamental theorem of algebra implies that the characteristic polynomial Δ(x)
has r (complex) roots, counting multiplicities. The next theorem describes how the solutions
look in the case of roots with multiplicity greater than one; for the proof see any book on
elementary combinatorics (for example, [230]).
Theorem 2.17 Let ξ ∈ C be any root of the characteristic polynomial Δ(x) of the recur-
rence relation (2.2) with multiplicity d. Then the basic solutions n
i
ξ
n
, i =0, 1,...,d−