[egin{align}
aF(z) + bG(z) &= sum_n (af_n+bg_n)z^n
\
z^mG(z) &= sum_n g_{n-m}z^n, quad mge 0
\
frac{G(z) - g_0 - g_1z - cdots - g_{m-1}z^{m-1}}{z^m} &= sum_ng_{n+m}z^n, quad mge 0
\
G(cz) &= sum_n c^ng_nz^n
\
G'(z) &= sum_n(n+1)g_{n+1}z^n
\
zG'(z) &= sum_nng_nz^n
\
int_0^z G(t){
m d} t &= sum_{nge 1}frac 1ng_{n-1}z^n
\
F(z)G(z) &= sum_nleft(sum_k f_ng_{n-k}
ight)z^n
\
frac 1{1-z}G(z) &= sum_nleft(sum_{kle n}g_k
ight)z^n
end{align}
]
以上是处理生成函数的基本方法。
以下是常用的生成函数。
[egin{array}{|l|l|}
hline
生成函数 & 封闭形式
\
hline
sum_{nge 0} [n=0]z^n & 1
\
sum_{nge 0}[n=m]z^n & z^m
\
sum_{n ge 0} z^n & frac 1{1-z}
\
sum_{nge 0}(-1)^nz^n & frac 1{1+z}
\
sum_{nge 0}[2mid n]z^n & frac 1{1-z^2}
\
sum_{nge 0}[mmid n]z^n & frac 1{1-z^m}
\
sum_{nge 0}(n+1)z^n & frac 1{(1-z)^2}
\
sum_{nge 0}2^nz^n & frac 1{1-2z}
\
sum_{nge 0}inom 4n z^n & (1+z)^4
\
sum_{nge 0}inom cn z^n & (1+z)^c
\
sum_{nge 0}inom {c+n-1}{n} z^n & frac 1{(1-z)^c}
\
sum_{nge 0}c^nz^n & frac 1{1-cz}
\
sum_{nge 0}inom{m+n}mz^n & frac 1{(1-z)^{m+1}}
\
sum_{nge 1}frac 1n z^n & ln frac 1{1-z}
\
sum_{nge 1}frac{(-1)^{n+1}}nz^n & ln (1+z)
\
sum_{nge 0}frac 1{n!}z^n & exp z
\
hline
end{array}
]
[G(z)+G(-z) = sum_ng_n(1+(-1)^n)z^n = 2sum_n[n 是偶数]g_nz^n
]
类似地,也可以只取奇数, 这对任意生成函数都可以使用。