Generating functions study notes

Continuous update

Generating function

Binomial coefficients (number of combinations)

\(\displaystyle C_n^m=\frac{n!}{m!(n-m)!}\)

\ (n \ in R, m \ in N \) Generalized binomial theorem coefficient

\(\displaystyle C_n^m=\frac{n(n-1)...(n-m+1)}{m!}\)

Generalized binomial theorem (a not necessarily integer)

\(\displaystyle (1+x)^a=\sum_{i=0}^{\infty}C_a^ix^i\)

About binomial identities

\(C_n^m=C_n^{n-m}\)

Polynomial derivation

\(\displaystyle (x^a)'=ax^{a-1}\)

Polynomials

\(\displaystyle \int x^adx=\frac{x^{a+1}}{a+1}\)

Usually generating function (of OGF) (reproduced hanging bell)
\ [\ DisplayStyle \ {n-sum_ \ geq0} [= n-m] = X ^ X ^ n-m \\ \ DisplayStyle \ {n-sum_ \ geq0} n-X ^ = \ frac {1} {1-x } \\ \ displaystyle \ sum_ {n \ geq0} ^ mx ^ n = \ frac {1-x ^ {m + 1}} {1-x} \\ \ displaystyle \ sum_ { n \ geq m} x ^ m = \ frac {x ^ m} {1-x} \\ \ displaystyle \ sum_ {n \ geq0} c ^ nx ^ n = \ frac {1} {1-cx} \\ \ displaystyle \ sum_ {n \ geq0 } C_ {n + k-1} ^ {n} x ^ n = \ frac {1} {(1-x) ^ k} \\ \ displaystyle \ sum_ {n \ geq0} x ^ {nk} = \ frac {1} {1-x ^ k} \\ \ displaystyle \ sum_ {n \ geq0} \ frac {c ^ nx ^ n} {n!} = e ^ {cx} \\ \ displaystyle \ sum_ {n \ geq0 } \ frac {(- 1) ^ {n-1}} {n} x ^ n = ln (1 + x) \\ \ displaystyle \ sum_ {n \ geq0} \ frac { 1} {n} x ^ n
= ln \ frac {1} {1-x} \] prefix and the generating function multiplied by \ (\ displaystyle \ sum_ {i = 0} ^ {\ infty} x ^ i = (1 -x) ^ {- 1} \ )

Multiplied by the difference generating function \ ((1-x) \ )

Exponential generating function (EGF)

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Origin www.cnblogs.com/wljss/p/12122640.html