A note on a conjecture of Borwein and Choi

A polynomial P(X) with coefficients {ǃ} of odd degree N - 1 is cyclotomic if and only if¶¶P(X) = ±F_p1 (±X)F_p2(±X^p1) ?F_pr(±X^{p1 p2 ?p_r-1}) P(X) = \pm \Phi_{p1} (\pm X)\Phi_{p2}(\pm X^{p1}) \cdots \Phi_{p_r}(\pm X^{p1 p2 \cdots p_r-1}) ¶where N = p₁ p₂ · · · p_r and the p_i are primes, not necessarily distinct, and where F_p(X) : = (X^p - 1) / (X - 1) \Phi_{p}(X) := (X^{p} - 1) / (X - 1) is the p-th cyclotomic polynomial. This is a conjecture of Borwein and Choi 1]. We prove this conjecture for a class of polynomials of degree N - 1 = 2^r p^l - 1 N - 1 = 2^{r} p^{\ell} - 1 for any odd prime p and for integers r, l\geqq 1 r, \ell \geqq 1 .