离散微积分积分表

默认 \(n\) 为常数，\(x\) 为自变量。

省略不定求和的 \(\delta x\) 和 \(+C\)。

MMA 缺省源

DD = DifferenceDelta;
FP = FactorialPower;
H = HarmonicNumber;
Binom = Binomial;

幂

（前提条件为 \(n \ne 1\)，\(n = 1\) 时平凡）

\[\sum n^x = \dfrac {n^x} {n-1} \]

\[\Delta \left( n^x \right) = (n-1) n^x \]

下降幂

（前提条件为 \(n \ne -1\)，\(n = -1\) 时见调和数部分）

\[\sum x^{\underline n} = \dfrac {x^{\underline {n+1}}} {n+1} \]

\[\Delta \left( x^{\underline n} \right) = n x^{\underline{n-1}} \]

\[\Delta \left( n^{\underline x} \right) = (n-x-1) n^{\underline x} \]

exp

\[\sum 2^x = 2^x \]

\[\Delta \left( 2^x \right) = 2^x \]

二项式系数

\[\sum \binom x n = \binom x {n+1} \]

\[\Delta \binom x n = \binom x {n-1} \]

更一般地：

\[\sum \binom {x+m} n = \binom {x+m} {n+1} \]

\[\Delta \binom {x+m} n = \binom {x+m} {n-1} \]

注：遇到组合数相关可以尝试 \(\dbinom x n = \dfrac {x^{\underline n}} {n!}\)。

调和数

\[\sum x^{\underline{-1}} = H_x \]

\[\sum H_x = xH_x - x \]

\[\sum x H_x = \dfrac {x(x-1)} 2 H_x - \dfrac {x(x-1)} 4 \]