It is closely related to the Euler transform, which is the result of applying the binomial transform to the sequence associated with its ordinary generating function.
The binomial transform, T, of a sequence, {an}, is the sequence {sn} defined by Formally, one may write for the transformation, where T is an infinite-dimensional operator with matrix elements Tnk.
The original series can be regained by The binomial transform of a sequence is just the nth forward differences of the sequence, with odd differences carrying a negative sign, namely: where Δ is the forward difference operator.
This is standard usage for example in On-Line Encyclopedia of Integer Sequences.
Both versions of the binomial transform appear in difference tables.
(The n-th number in the m-th line is am,n = 3n−2(2m+1n2 + 2m(1+6m)n + 2m-19m2), and the difference equation am+1,n = am,n+1 - am,n holds.)
The top line read from left to right is {an} = 0, 1, 10, 63, 324, 1485, ...
The top line read from right to left is {bn} = 1485, 324, 63, 10, 1, 0, ...
The transform connects the generating functions associated with the series.
For the ordinary generating function, let and then The relationship between the ordinary generating functions is sometimes called the Euler transform.
It commonly makes its appearance in one of two different ways.
In one form, it is used to accelerate the convergence of an alternating series.
That is, one has the identity which is obtained by substituting x = 1/2 into the last formula above.
The terms on the right hand side typically become much smaller, much more rapidly, thus allowing rapid numerical summation.
Here, the Euler transform takes the form: [See [1] for generalizations to other hypergeometric series.]
The binomial transform, and its variation as the Euler transform, is notable for its connection to the continued fraction representation of a number.
have the continued fraction representation then and For the exponential generating function, let and then The Borel transform will convert the ordinary generating function to the exponential generating function.
Knuth and O. Patashnik: Concrete Mathematics: A Foundation for Computer Science, Addison-Wesley (1989).
It is easy to see that the binomial convolution is associative and commutative, and the sequence
serves as the identity under the binomial convolution.
forms an Abelian group under the binomial convolution.
The binomial convolution arises naturally from the product of the exponential generating functions.
There is also another binomial convolution in the mathematical literature.
The binomial convolution of arithmetical functions
is the canonical factorization of a positive integer
This convolution appears in the book by P. J. McCarthy (1986) and was further studied by L. Toth and P. Haukkanen (2009).
When the sequence can be interpolated by a complex analytic function, then the binomial transform of the sequence can be represented by means of a Nörlund–Rice integral on the interpolating function.
Prodinger gives a related, modular-like transformation: letting gives where U and B are the ordinary generating functions associated with the series
In the case where the binomial transform is defined as Let this be equal to the function
, 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