## Factorial ratios that are integers.(English)Zbl 1042.11511

Let $$m,k$$ and $$j$$ be integers with $$m\geq k>j\geq 0$$ and let $$Q(j,m,k)=\prod^j_{i=0}\gcd(m-i, \text{lcm}(k,k-1,\cdots,k-i))$$. The author gives divisibility properties of $$Q$$. He proves:
Theorem 1. $$Q(j,m,k)(m-j+1)!/k!(m-k)!$$ is an integer. Theorem 2. For integers $$s\geq 1$$, $$r\geq 0$$ and $$n\geq 1$$, the number $$(2r+s)!/r!(s-1)!Q(r,r+s+2n,r+s+n)$$ is an integer.

### MSC:

 11B65 Binomial coefficients; factorials; $$q$$-identities 05A10 Factorials, binomial coefficients, combinatorial functions
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