The *binomial transform* of a sequence is the sequence satisfying

In this post we prove the following:

*Theorem*: If is the binomial transform of , then

In other words, the *m*th finite difference of is the binomial transform of the shifted sequence . This theorem is useful because once we know one binomial coefficient identity in the form of a binomial transform we can use it to generate others.

*Proof*: First, we have

In the second-to-last step we use Pascal’s recurrence plus the fact that . In the last step we use .

Therefore, and, in general,

*Example*: Start with the identity (I’ve proved this identity three different ways so far on this blog: here, here, and here.) Multiplying it by -1, we have

Applying the theorem we just proved yields

Since , this simplifies to

This identity can be generalized by taking finite differences. I’ll leave it to the reader to prove the generalization

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