基于三维递推算法的牛顿二项式定理研究

Newton’s Binomial Theorem Based on Three-Dimensional Recursive Algorithms

摘要: 牛顿二项式展开的递推算法先不计算(x + y)k−1而直接计算(x + y)k，其中指数项k是“二维”有理数(分子和分母各一维)或负数。针对牛顿二项式展开项系数的计算问题，本文给出了一种分拆函数有三个变量的新的“三维”递推算法。该方法把逐步递推的过程转化成了整数(或有理数)的分拆计算进而可直接求出任意展开项的系数，同时也证明和解释了牛顿二项式展开项系数的本质是分拆函数的比值。另一方面，当k不是正整数时，牛顿二项展开式给出的多项式环都是无穷级数的形式。而本文给出的展开式中除了有无穷级数形式的多项式环以外，还存在有限序列和形式的多项式环。因此，新算法可以用来进一步研究高斯整数环中的素元。

Abstract: The recursive algorithm of Newton’s binomial expansion calculates directly (x + y)k−1 without calculating (x + y)k, where the exponential term k is a “two-dimensional” rational number (one-dimensional molecule and denominator) or a negative number. To solve the problem of calculating the coefficients of Newton’s binomial expansion terms, this paper presents a new “three-dimensional” recursive algorithm for splitting functions with three variables. This method transforms the step-by-step recursive process into the splitting calculation of integers (or rational numbers) and then directly calculates the coefficients of arbitrary expansion terms. At the same time, it proves and explains that the essence of the coefficients of Newton’s binomial expansion terms is the ratio of splitting functions. On the other hand, when k is not a positive integer, the polynomial rings given by Newton’s binomial expansion are all in the form of infinite series. In addition to polynomial rings in infinite series form, there are polynomial rings in finite sequence and form in the expansions given in this paper. Therefore, the new algorithm can be used to further study the prime elements in the Gauss integer ring.