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 \(\omega_p(n)= \sum_{k\geq 0} [{n\over(p- 1)p^k}]\).


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


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[1] Bhargava, M., The factorial function ⋯ and generalizations, Amer. math. monthly, 107, 783-799, (2000) · Zbl 0987.05003
[2] Chabert, J.-L., Une caractérisation des polynômes prenant des valeurs entières sur tous LES nombres premiers, Canad. math. bull., 39, 402-407, (1996) · Zbl 0885.11024
[3] Chabert, J.-L.; Chapman, S.; Smith, W., A basis for the ring of polynomials integer-valued on prime numbers, (), 271-284 · Zbl 0967.13015
[4] Sloane, N.J.A., The on-line encyclopedia in integer sequences · Zbl 1274.11001
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