## The factorial function and generalizations.(English)Zbl 0987.05003

Embarking from some number-theoretic appearances of the factorial function the author aims to generalize this function as well as some results in which it plays a crucial role. In order to achieve the goal, he introduces the notion of a $$p$$-ordering: Let $$S$$ be an arbitrary subset of the integers $${\mathbb{Z}}$$, and fix a prime $$p$$. A $$p$$-ordering of $$S$$ is a sequence $$\{ a_i \}_{i=0}^\infty$$ of elements of $$S$$ such that $$a_0 \in S$$ is chosen first and then for $$i \in {\mathbb{N}}$$ in increasing order the element $$a_i \in S$$ is chosen so that it minimizes the highest power of $$p$$ dividing the product $$(a_i - a_0)(a_i - a_1)\cdots (a_i - a_{i-1})$$.
For $$z \in {\mathbb{Z}}$$ let $$w_p(z)$$ denote the highest $$p$$-power dividing $$z$$. Then, for any $$p$$-ordering $$\{ a_i \}_{i=0}^\infty$$ of $$S$$ we define the associated $$p$$-sequence of $$S$$ as $$\{ \nu_k(S,p) \}$$ where $$\nu_k(S,p) := w_p( (a_k - a_0)(a_k - a_1)\cdots (a_k - a_{k-1}))$$. The notation is justified by
Theorem 5. The associated $$p$$-sequence of $$S$$ is independent of the choice of the $$p$$-ordering of $$S$$.
Now define the factorial function of $$S$$ by $$k!_S := \prod_{p} \nu_k(S,p)$$ and note that in particular $$k!_{\mathbb{Z}} = k!$$.
Revising some old theorems the author then proves the following generalized versions.
Theorem 8. For any nonnegative integers $$k$$ and $$l$$, the number $$(k + l)!_S$$ is a multiple of $$k!_S \cdot l!_S$$.
Theorem 9. Let $$f$$ be a primitive polynomial of degree $$k$$, and let $$d(S,f):= \text{gcd}\{ f(a): a\in S\}$$. Then $$d(S,f)$$ divides $$k!_S$$.
Theorem 10. Let $$a_0,a_1,\dots, a_n$$ be any $$n+1$$ integers of $$S$$. Then the product $$\prod_{i < j} (a_i - a_j)$$ is a multiple of $$0!_S 1!_S \dots n!_S$$.
Theorem 11. The number of polynomial functions from $$S$$ to $${\mathbb{Z}}/n{\mathbb{Z}}$$ is given by $$\displaystyle\prod_{k=0}^{n-1}n/ \text{gcd}(n,k!_S)$$.
Theorem 23. A polynomial is integer-valued on a subset $$S$$ of $${\mathbb{Z}}$$ if and only if it can be written as a $${\mathbb{Z}}$$-linear combination of the polynomials $(x - a_{0,k})(x - a_{1,k}) \cdots(x - a_{k-1,k})/ k!_S, \quad k = 0, 1, 2,\dots,$ where $$\{ a_{i,k} \}_{i=0}^\infty$$ is a sequence in $${\mathbb{Z}}$$ which for each prime $$p$$ dividing $$k!_S$$ is termwise congruent modulo $$\nu_k(S,p)$$ to some $$p$$-ordering of $$S$$.
Many of the results can be generalized further to subsets $$S$$ of Dedekind rings $$R$$, as is demonstrated in
Theorem 24. The set of polynomials that are $$R$$-valued on a subset $$S$$ of a Dedekind domain $$R$$ has a regular basis if and only if $$k!_S$$ is a principal ideal for all $$k \geq 0$$. If this is the case, then a regular basis may be given as in Theorem 23.
Thus, this fundamental problem about integer-valued polynomials, first put forth by Pólya in 1919 has now been resolved.
Before the paper concludes with a series of interesting questions the author points out some applications to interpolation problems for which the interested reader may consult the paper.

### MSC:

 05A10 Factorials, binomial coefficients, combinatorial functions 11B65 Binomial coefficients; factorials; $$q$$-identities 33B99 Elementary classical functions 11C08 Polynomials in number theory

### Keywords:

factorial function; generalized factorials; $$p$$-ordering
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