生成函數是利用冪級數中變數的係數來表達數列，在組合數學和機率論上都有廣泛的應用。本文中將介紹四種單變數的生成函數重要的性質：一般生成函數、指數生成函數、機率生成函數和動差生成函數，並透過大量的例子說明他們在組合數學和機率上的應用，並討論一些數學競賽中與生成函數相關的問題。

Generating functions express a sequence as coefficients arising from a power series in variables. They have many applications in combinatorics and probability. In this paper, we will investigate the important properties of four kinds of generating functions in one variables: ordinary generating unction, exponential generating function, probability generating function and moment generating function. Many examples with applications in combinatorics and probability, will be discussed. Finally, some
well-known contest problems related to generating functions will be addressed.

致謝 i
中文摘要 ii
英文摘要 iii
1 前言 1
2 一般生成函數(ordinary generating function) 2
2.1 基本性質 2
2.2 計數8
2.3 整數分割 11
2.4 遞迴關係式 13
2.5 數列和與恆等式 16
3 指數生成函數(exponential generating function) 22
3.1 基本性質 22
3.2 排列 25
4 機率生成函數(probability generating function) 30
4.1 基本性質 30
4.2 期望值和變異數 31
4.3 獨立隨機變數的和 36
4.4 混合分佈 39
5 動差生成函數(moment generating function) 41
5.1 基本性質 42
5.2 期望值和變異數 45
5.3 刻畫分佈 47
5.4 極限逼近 50
6 數學競賽中的生成函數 54
附錄A：使用Mathematica計算生成函數 60
A.1 一般生成函數、指數生成函數和機率生成函數 60
A.2 期望值、變異數與動差生成函數 63
附錄B：縮寫與符號 65
B.1 數學競賽名稱縮寫 65
B.2 符號 66
附錄C：習題解答提示 66
附錄D：重點整理 66
D.1: 2 一般生成函數 66
D.2: 3 指數生成函數 68
D.3: 4 機率生成函數 69
D.4: 5 動差生成函數 70
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