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:

[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, (Factorization in Integral Domains. Factorization in Integral Domains, Lecture Notes in Pure and Appl. Math., vol. 189 (1997), Dekker: Dekker New York), 271-284 · Zbl 0967.13015
[4]	Sloane, N. J.A., The On-Line Encyclopedia in Integer Sequences · Zbl 1274.11001