  The main object of study in this paper is the completion Z[q]^N=\varprojlim_n
Z[q]/((1-q)(1-q^2)...(1-q^n)) of the polynomial ring Z[q], which arises from
the study of a new invariant of integral homology 3-spheres with values in
Z[q]^N announced by the author, which unifies all the sl_2
Witten-Reshetikhin-Turaev invariants at various roots of unity. We show that
any element of Z[q]^N is uniquely determined by its power series expansion in
q-\zeta for each root \zeta of unity. We also show that any element of Z[q]^N
is uniquely determined by its values at the roots of unity. These results may
be interpreted that Z[q]^N behaves like a ring of ``holomorphic functions
defined on the set of the roots of unity''. We will also study the
generalizations of Z[q]^N, which are completions of the polynomial ring R[q]
over a commutative ring R with unit with respect to the linear topologies
defined by the principal ideals generated by products of powers of cyclotomic
polynomials.
