Representations by ordered Bell and degenerate ordered Bell polynomials

The ordered Bell numbers (also called Fubini numbers) bn arise from number theory and various counting problems in enumerative combinatorics (see [5,14]). The ordered Bell numbers bn appeared already in 1859 work of Cayley [3], who used them to count certain plane trees with n + 1 totally ordered leaves. While the (unordered) Bell numbers Beln given by e (e−1) = ∑∞ n=0Beln t n! count partitions of [n] = {1, 2, . . . , n} into nonempty disjoint subsets, the ordered Bell numbers count ordered partitions of [n]. Equivalently, the ordered Bell numbers bn count either the number of weak orderings on a set of n elements or the mappings from [n] to itself whose image is [l], 1 ≤ l ≤ n. They also count formulas in Fubini’s theorem when rearranging the order of summation in multiple sums. We let the reader refer to [19], for details on the numerous uses of the ordered Bell numbers in counting problems. Let p(x) ∈ C[x], with deg p(x) = n. Write p(x) = ∑n k=0 akBk(x), where Bn(x) are the Bernoulli polynomials (see (1.3)). Then it is known (see [12]) that