Binomial Transform - Generalizations

Generalizations

Prodinger gives a related, modular-like transformation: letting

gives

where U and B are the ordinary generating functions associated with the series and, respectively.

The rising k-binomial transform is sometimes defined as

The falling k-binomial transform is

.

Both are homomorphisms of the kernel of the Hankel transform of a series.

In the case where the binomial transform is defined as

Let this be equal to the function

If a new forward difference table is made and the first elements from each row of this table are taken to form a new sequence, then the second binomial transform of the original sequence is,

If the same process is repeated k times, then it follows that,

Its inverse is,

This can be generalized as,

where is the shift operator.

Its inverse is