关于生成函数, 以下直接引用 wiki 百科上的介绍:
In mathematics, a generating function is a way of encoding an infinite sequence of numbers ($a_n$) by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence.
Unlike an ordinary series, the formal power series is not required to converge: in fact, the generating function is not actually regarded as a function, and the “variable” remains an indeterminate. Generating functions were first introduced by Abraham de Moivre in 1730, in order to solve the general linear recurrence problem. One can generalize to formal power series in more than one indeterminate, to encode information about infinite multi-dimensional arrays of numbers.
正如这段介绍中所说, 生成函数就是描述数列的另一种不同的方法而已, 这种方法将整个序列视作了一个对象进行考虑, 更具体的说, 就是个幂级数, 其系数有着某些特定含义.
作为一篇小品, 本文只简单介绍一下生成函数定义, 并利用该方法研究斐波那契数列.