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**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.

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.

Reviewer: Wolfgang Lempken (Essen)