由量子整数乘法产生的形式幂级数
Formal power series arising from multiplication of quantum integers
摘要：
对于量子整数[n]_q = 1+q+q^2+. + q^{n-1}存在自然多项式乘法使得[mn]_q = [m]_q \otimes_q [n]_q。这个乘法由函数方程f_{mn}（q）= f_m（q）f_n（q^m）给出，定义在多项式序列{f_n（q）}上，使得对于所有n，f_n（0）=1。证明了如果{fn（q）}是该函数方程的解，则序列{fn（q）}收敛于形式幂级数F（q）.量子乘法还导致函数方程f（q）F（q^m）= F（q），其中f（q）是固定多项式或具有常数项f（0）=1的形式幂级数，并且F（q）=1+ \sum_{k=1}^{\infty} b_kq^k是形式幂级数。证明了该函数方程对每一个多项式或形式幂级数f（q）都有唯一解F（q）。若f（q）的次数至多为m-1，则F（q）的系数b_k可用f（q）的系数和k的m-adic表示来表示.本文还对系数在任意域或整数域上的形式幂级数级数的收敛性质进行了讨论。
英文摘要：
For the quantum integer [n]_q = 1+q+q^2+... + q^{n-1} there is a natural polynomial multiplication such that [mn]_q = [m]_q \otimes_q [n]_q. This multiplication is given by the functional equation f_{mn}(q) = f_m(q) f_n(q^m), defined on a sequence {f_n(q)} of polynomials such that f_n(0)=1 for all n. It is proved that if {f_n(q)} is a solution of this functional equation, then the sequence {f_n(q)} converges to a formal power series F(q). Quantum mulitplication also leads to the functional equation f(q)F(q^m) = F(q), where f(q) is a fixed polynomial or formal power series with constant term f(0)=1, and F(q)=1+\sum_{k=1}^{\infty}b_kq^k is a formal power series. It is proved that this functional equation has a unique solution F(q) for every polynomial or formal power series f(q). If the degree of f(q)is at most m-1, then there is an explicit formula for the coefficients b_k of F(q) in terms of the coefficients of f(q) and the m-adic representation of k. The paper also contains a review of convergence properties of formal power series with coefficients in an arbitrary field or integeral domain.
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