## Integer-valued polynomials on prime numbers and logarithm power expansion.(English)Zbl 1120.11016

The coefficient $$A_n(m)$$ of $$x^n$$ in the power series expansion $\Biggl(\sum^\infty_{k=1} {x^k\over k+1}\Biggr)^m= \sum^\infty_{n=0} A_n(m) x^n$ has the form $$B_n(m)/d(n)$$, where $$B_n(m)$$ is a primitive polynomial in $$\mathbb{Z}[m]$$. It is shown that the denominator $$d(n)$$ is an integer given by the following product extended over primes $$p$$: $d(n)= \prod_{p\leq n+1} p^{\omega_p(n)},$ where $$\omega_p(n)= \sum_{k\geq 0} [{n\over(p- 1)p^k}]$$.

### MSC:

 11C08 Polynomials in number theory 11B83 Special sequences and polynomials 11B65 Binomial coefficients; factorials; $$q$$-identities

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### References:

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