加法表示函数的若干性质

Some Properties of Additive Representation Functions

令A={a_1,a_2,…}(a_1≤a_2≤…)是一个无限非负整数序列.设k≥2是固定的正整数,对n∈N,令R_k(A,n)表示方程a_i_1+…+a_i_k=n解的个数.令R_k^((1))(A,n)及R_k^((2))(A,n)分别表示上述方程带限制条件i_1<…<i_k及i_1≤…≤i_k时解的个数.最近,陈永高和本文作者证明了如下结果:设d是一个正整数,若对充分大的所有n皆有R_k(A,n)≥d,则R_k(A,n)≥d+2[k/2]!d^(1/2)+([k/2]!)~2对无穷多个n成立.本文获得了R_k^((1))(A,n)及R_k^((2))(A,n)的相关结果.

Let A = {a_1,a_2...}(a_1≤a_2≤…) be an infinite sequence of nonnegative integers.Let k > 2 be a fixed integer and for n ∈ N,let R_k(A,n) be the number of solutions of a_o_1 +… + a_i_k = n.Let R_k^((1))(A,n) and R_2^((2))(A,n) denote the number of solutions with the additional restrictions i_1<…<i_k,and i_1 < …≤i_k respectively.Recently,Yong-Gao Chen and the author proved the following result:let d be a positive integer.If Rk(n) > d for all sufficiently large integers n,then R_k(n)≥d + 2[k/2]!d^1/2+([k/2]!)~2 for infinitely many positive integers n.In this paper,we obtain the analogue results for R_k^((1))(A,n) and R_2^((2))(A,n).