An interesting way of approaching this might be via the p-adic Gamma function of Morita. It's basically a slightly fixed up version of the ordinary Gamma function: Instead of always using the recursion Gamma(x+1) = x Gamma(x), make the recursion Gamma(x+1) = Gamma(x) if x is an integer divisible by p. The amazing thing is that the resulting function is a p-adic analytic function (since the integers are dense in the p-adic integers, it extends uniquely to a p-adic continuous function), and whose value is always a p-adic unit, so it can be unambiguously reduced modulo. You can then use that to make a p-adic beta function (the beta function being the right generalization of the binomial coefficient). Victor On Sat, Dec 5, 2009 at 3:11 PM, Edwin Clark <eclark@math.usf.edu> wrote:
On Sat, 5 Dec 2009, Eugene Salamin wrote:
Definition: A polynomial p(x) with rational coefficients is said to be "almost integer" if p(n) is an integer for all integers n. Example: x(x+1)/2.
It appears that such polynomials have been considered previously under the name "integer polynomials". It is interesting that they form a free abelian group with basis: t(t - 1)...(t - k + 1)/k!, k = 0,1,2,... according to:
http://en.wikipedia.org/wiki/Integer-valued_polynomial
I encounted them recently when I learned of "Schinzel's hypothesis H"
http://en.wikipedia.org/wiki/Schinzel's_hypothesis_H.
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