updd 2021-02-25 刚学会怎么把html传到github然后弄出来一个url让你访问
https://yhm138.github.io/personal_yhm138/memos/gf.html
普通生成函数OGF
\[f(x)=\sum_{n=0}^{\infty}a_nx^n \]指数生成函数 EGF
\[f(x)=\sum_{n=0}^{\infty}\frac{a_n}{n!}x^n \]Dirichlet生成函数
\[f(s)=\sum_{n=1}^{\infty}\frac{a_n}{n^s} \]Notation
\(P()\) denotes Polynomial
\(S_1(n,k)\) denotes the Stirling's number of first kind，and \(S_2(n,k)\) so on
\(\mu(n)\) denotes mobius function
\(p\) prime
OGFOGF property
\[f(x)g(x)\stackrel{}{\longleftrightarrow}\{\sum_{n=0}^{\infty}a_kb_{n-k} \}_{n=0}^{\infty} \] \[f^k(x)\stackrel{}{\longleftrightarrow} \{ \sum_{n_1+n_2+...+n_k=n}a_{n_1}a_{n_2}a_{n_3}...a_{n_k} \}_{n=0}^{\infty} \] \[\frac{f(x)}{1-x}\stackrel{}{\longleftrightarrow} \{ \sum_{j=0}^n a_j \}_{n=0}^{\infty} \] \[P(xD)f\stackrel{}{\longleftrightarrow} \{ P(n)a_n \}_{n=0}^{\infty} \]some OGF instances
\[\frac{1}{1-x}{\longleftrightarrow} \{ \ 1\ \}_{n=0}^{\infty} \] \[\frac{x}{(1-x)^2}{\longleftrightarrow} \{ \ n\ \}_{n=0}^{\infty} \] \[\frac{1}{(1-x)^k}{\longleftrightarrow} \{ \ \tbinom{n+k-1}{n}\ \}_{n=0}^{\infty} \] \[\frac{1}{(1-rx)^k}{\longleftrightarrow} \{ \ \ \tbinom{n+k-1}{n}r^n\ \ \}_{n=0}^{\infty} \] \[\frac{1}{1-cx}{\longleftrightarrow} \{ c^n \}_{n=0}^{\infty} \] EGFEGF property
\[D^k f{\longleftrightarrow} \{ a_{n+k} \}_{n=0}^{\infty} \] \[xDf{\longleftrightarrow} \{ na_n \}_{n=0}^{\infty} \] \[P(xD)f {\longleftrightarrow} \{ P(n)a_n \}_{n=0}^{\infty} \] \[f(x)g(x){\longleftrightarrow} \{ \sum_{k=0}^n \tbinom{n}{k} a_kb_{n-k} \}_{n=0}^{\infty} \] \[f(x)g(x)h(x)={\longleftrightarrow} \{ \sum_{i+j+k=n\\i,j,k\geq0}\tbinom{n}{i,j,k}a_ib_jc_k \}_{n=0}^{\infty} \] \[f^k(x){\longleftrightarrow} \{ \sum_{n_1+n_2+...+n_k=n\\n_i\geq0,i=1,2,...,k}\tbinom{n}{n_1,n_2,...n_k}a_{n_1}a_{n_2}...a_{n_k} \}_{n=0}^{\infty} \]some EGF instances
\[e^x{\longleftrightarrow} \{ 1 \}_{n=0}^{\infty} \] \[e^{cx}{\longleftrightarrow} \{ c^n \}_{n=0}^{\infty} \] \[\frac{(e^x-1)^k}{k!}{\longleftrightarrow} \{ \ S_2(n,k)\ \}_{n=0}^{\infty} \] \[\frac{[ln(1+x)]^k}{k!}{\longleftrightarrow} \{ \ S_1(n,k)\ \}_{n=0}^{\infty} \] Dirichlet生成函数Dirichlet GF property
\[f(s)g(s){\longleftrightarrow} \{ \sum_{d|n}a_db_{\frac{n}{d}} \}_{n=1}^{\infty} \] \[f^k(s){\longleftrightarrow} \{ \sum_{n_1n_2...n_k=n}a_{n_1}a_{n_2}...a_{n_k} \}_{n=1}^{\infty} \]some Dirichlet GF instances
\[\zeta(s){\longleftrightarrow} \{ 1 \}_{n=1}^{\infty} \] \[[\zeta(s)]^2{\longleftrightarrow} \{ \sum_{d|n}1 \}_{n=1}^{\infty} \] \[\frac{1}{\zeta(s)}{\longleftrightarrow} \{ \ \mu(n)\ \}_{n=1}^{\infty} \] \[[\zeta(s)]^k{\longleftrightarrow} \{ n可分解为k个有序正因子积的方法数 \}_{n=1}^{\infty} \] \[[\zeta(s)-1]^k{\longleftrightarrow} \{ n可分解为k个非平凡有序正因子积的方法数 \}_{n=1}^{\infty} \] \[\prod_{p}(\sum_{k=0}^{\infty}f(p^k)p^{-ks}){\longleftrightarrow} \{ 积性数论函数f(n) \}_{n=1}^{\infty} \]先写到这，不定期更新
编辑公式不易，转载请注明出处
2020-08-20