structural:scholar_edition, structural:expansion
Generalizations

There are many notions of generalized Stirling numbers that may be defined (depending on application) in a number of differing combinatorial contexts. In so much as the Stirling numbers of the first kind correspond to the coefficients of the distinct polynomial expansions of the single factorial function, n ! = n (n − 1) (n − 2) ⋯ 2 ⋅ 1 {\displaystyle n!=n(n-1)(n-2)\cdots 2\cdot 1}, we may extend this notion to define triangular recurrence relations for more general classes of products.

In particular, for any fixed arithmetic function f: N → C {\displaystyle f:\mathbb {N} \rightarrow \mathbb {C} } and symbolic parameters x, t {\displaystyle x,t}, related generalized factorial products of the form

may be studied from the point of view of the classes of generalized Stirling numbers of the first kind defined by the following coefficients of the powers of x {\displaystyle x} in the expansions of (x) n, f, t {\displaystyle (x)_{n,f,t}} and then by the next corresponding triangular recurrence relation:

These coefficients satisfy a number of analogous properties to those for the Stirling numbers of the first kind as well as recurrence relations and functional equations related to the f-harmonic numbers, F n (r) (t):= ∑ k ≤ n t k / f (k) r {\displaystyle F_{n}^{(r)}(t):=\sum _{k\leq n}t^{k}/f(k)^{r}}.

One special case of these bracketed coefficients corresponding to t ≡ 1 {\displaystyle t\equiv 1} allows us to expand the multiple factorial, or multifactorial functions as polynomials in n {\displaystyle n}.

The Stirling numbers of both kinds, the binomial coefficients, and the first and second-order Eulerian numbers are all defined by special cases of a triangular super-recurrence of the form

for integers n, k ≥ 0 {\displaystyle n,k\geq 0} and where | n k | ≡ 0 {\displaystyle \left|{\begin{matrix}n\\k\end{matrix}}\right|\equiv 0} whenever n < 0 {\displaystyle n<0} or k < 0 {\displaystyle k<0}. In this sense, the form of the Stirling numbers of the first kind may also be generalized by this parameterized super-recurrence for fixed scalars α, β, γ, α ′, β ′, γ ′ {\displaystyle \alpha,\beta,\gamma,\alpha ^{\prime },\beta ^{\prime },\gamma ^{\prime }} (not all zero).