table of differences and sums

(A constant $C$ should be added to each sum.)

$\begin{align*} &\text{Difference}&&f(x) &&\text{Sum}& \\ &k\Delta U_x &&kU_x &&k\sum U_x & \\ &(a-1)a^x &&a^x &&\frac{a^x}{a-1}, a\neq 1 & \\ &n\fp{x}{n-1} &&\fp{x}{n} &&\frac{\fp{x}{n+1}}{n+1} & \\ &\frac{-n}{\fp{x}{n+1}} &&\fr{\fp{x}{n}} &&\fr{(1-n)\fp{x}{n-1}},n\neq 1 & \\ &bn\fp{(a+bx)}{n-1} &&\fp{(a+bx)}{n} &&\frac{\fp{(a+bx)}{n+1}}{b(n+1)} & \\ &\frac{-bn}{\fp{(a+bx)}{n+1}} &&\frac{1}{\fp{(a+bx)}{n}} && & \\ &\binom{x}{n-1} &&\binom{x}{n} &&\binom{x}{n+1} & \\ &U_x\Delta V_x+V_{x+1}\Delta U_x &&U_xV_x && & \\ &x(x!) &&x! && & \end{align*}$

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table of differences and sums

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