On coefficients of Carlitz cyclotomic polynomials

Let n∈Z+n∈Z+, and Φn(x)Φn(x) be the nth classical cyclotomic polynomial. In [4, Theorem 1], D. Lehmer showed that the geometric mean of {Φs(1):s,n∈Z+,s≤n}→e≈2.71828{Φs(1):s,n∈Z+,s≤n}→e≈2.71828, as n→∞n→∞. Replacing ZZ by Fq[T]Fq[T], and the n  th elementary cylcotomic polynomial Φn(x)Φn(x) by the Carlitz m  -cyclotomic polynomial Φm(x)Φm(x), where m∈Fq[T]m∈Fq[T], we obtain an analogue to Lehmer's result. We also express Φm(0)∈F2[T]Φm(0)∈F2[T] in terms of ϕ⁎(⋅)ϕ⁎(⋅), the Pillai polynomial function. The resulting expression is a function field analogue of Hölder's formula for Φn(1)Φn(1).