此帖非常重要，所以我写了中英对照版本。
This post is really important so we have Chinese and English version.
我是专门研究代数的数学家。
当我第一次用差分算子求出方幂和的时候，我感到惊讶。
方幂和有公式，那个公式有伯努利数。
不过从朱世杰恒等式可以看到这类求和可以有更简单的表达式。
其实我们只要用牛顿级数将多项式写成组合数，就可以求出方幂和。
同样地，当我们想求出差比数列的和时，
可能打算从等比数列递推上去，我们根本没有公式。
我后悔学了这么久才知道有差分算子。
我只是想大家将来能够更早知道这件事。
这是一个被定义成f(x+1)-f(x)的算子的故事。
我认为在学微分之前就应该先学差分。
可惜我学完微分之后还没有听说过有这种东西。
I am a mathematician interested in algebra though I'm just working as an accountant in Hong Kong.
I am surprised that Finite Difference has become an excellent tool on power sum.
While Faulhaber's formula is well-known as a general formula to this kind of problem, we found that the Hockey-stick identity is able to deal with some kind of power sum with a very simple formula.
We can rewrite a polynomial to binomial coefficient with Newton's series, so the power sum can be expressed by a much clear form with Finite Difference.
Again, when we're going to find the sum of arithmetic–geometric sequence, we're just expected to derive it from the sum of geometric sequence.
We don't even know a general formula for this problem.
I regret that I learn the Finite Difference so late.
I just hope that everyone would have the chance to learn this earlier.
Here's a story with something defined as f(x+1)-f(x).
It's a Linear operator which is something we should study before differentiation.
And I still haven't heard of it after studying differentiation.
详情可见条目求和符号及维基教科书代数/本书课文/求和
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