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You may be also interested in this formula for indefinite sum of $f(x)$:

$$\sum_x f(x)=-\sum_{k=1}^{\infty}\frac{\Delta^{k-1}f(x)}{k!}(-x)_k+C$$

where $(x)_k=\frac{\Gamma(x+1)}{\Gamma(x-n+1)} (x)_k=\frac{\Gamma(x+1)}{\Gamma(x-k+1)}$ is a falling factorial.

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You may be also interested in this formula for indefinite sum of $f(x)$:

$$\sum_x f(x)=-\sum_{k=1}^{\infty}\frac{\Delta^{k-1}f(x)}{k!}(-x)_k+C$$

where $(x)_k=\frac{x!}{(x-n)!} (x)_k=\frac{\Gamma(x+1)}{\Gamma(x-n+1)}$ is a falling factorial.

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You may be also interested in this formula for indefinite sum of $f(x)$:

$$\sum_x f(x)=-\sum_{k=1}^{\infty}\frac{\Delta^{k-1}f(x)}{k!}(-x)_k+C$$

where $(x)_k=\frac{x!}{(x-n)!}$ is a falling factorial.