Binomial Transform - Euler Transform

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

$\sum_{n=0}^\infty (-1)^n a_n = \sum_{n=0}^\infty (-1)^n \frac {\Delta^n a_0} {2^{n+1}}$

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.

The Euler transform can be generalized (Borisov B. and Shkodrov V., 2007):

$\sum_{n=0}^\infty (-1)^n {n+p\choose n} a_n = \sum_{n=0}^\infty (-1)^n {n+p\choose n}\frac {\Delta^n a_0} {2^{n+p+1}}$ ,

where p = 0, 1, 2,...

The Euler transform is also frequently applied to the Euler hypergeometric integral . Here, the Euler transform takes the form:

The binomial transform, and its variation as the Euler transform, is notable for its connection to the continued fraction representation of a number. Let have the continued fraction representation

then

and

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