从微积分的观点看高阶等差数列的求和

Summation of Arithmetic Progression of Higher Order in View of Calculus

通过差分算子给出了高阶等差数列的定义,并以朱世杰恒等式和朱世杰招差公式为工具解决了高阶等差数列的求和,强调了这一问题与普通的无限微积分中Newton-Leibniz公式求定积分这个标准问题之间的类似.此外,应用朱世杰招差公式给出了整数值多项式的经典刻划.

This expository paper provides a precise definition of arithmetic progression of higher order via the difference operator. The classical problem of the summation of arithmetic progression of higher order is solved by using two formulas due to Zhu Shijie.. Zhu＇s identity and Zhu＇s difference formula. The analogy between this discrete problem and the standard problem of calculating definite integral via Newton--Leib- niz formula in ordinary infinite Calculus is stressed. In addition, it characterizes an integer--valued polynomial via Zhu＇s difference formula.