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

Chabert, Jean-Luc

Eur. J. Comb. 28, No. 3, 754-761 (2007).

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

Reviewer: Tom M. Apostol (Pasadena)

MSC:

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

Software:

OEIS

References:

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