There are (at least) three types of Euler transforms (or transformations). The first is a set of transformations of hypergeometric functions, called Euler's hypergeometric transformations.
The second type of Euler transform is a technique for series convergence improvement which takes a convergent alternating series
 |
(1)
|
into a series with more rapid convergence to the same value to
 |
(2)
|
where the forward difference is defined by
 |
(3)
|
(Abramowitz and Stegun 1972; Beeler et al. 1972). Euler's hypergeometric and convergence improvement transformations are related by the
fact that when 
is taken in the second of Euler's
hypergeometric transformations
 |
(4)
|
where 
is a hypergeometric function, it gives
Euler's convergence improvement transformation of the series 
(Abramowitz and Stegun 1972, p. 555).
The third type of Euler transform is a relationship between certain types of integer sequences (Sloane and Plouffe 1995, pp. 20-21). If 
, 
, ... and 
, 
, ... are related by
 |
(5)
|
or, in terms of generating functions 
and 
,
![1+B(x)=exp[sum_(k=1)^infty(A(x^k))/k],](/images/equations/EulerTransform/NumberedEquation6.svg) |
(6)
|
then 
is said to be the Euler transform of 
(Sloane and Plouffe 1995, p. 20). The Euler transform
can be effected by introducing the intermediate series 
, 
, ... given by
 |
(7)
|
then
![b_n=1/n[c_n+sum_(k=1)^(n-1)c_kb_(n-k)],](/images/equations/EulerTransform/NumberedEquation8.svg) |
(8)
|
with 
.
Similarly, the inverse transform can be effected by computing the intermediate series
as
 |
(9)
|
then
 |
(10)
|
where 
is the Möbius function.
In graph theory, if 
is the number of unlabeled connected graphs on 
nodes satisfying some property, then 
is the total number of unlabeled
graphs (connected or not) with the same property. This application of the Euler
transform is called Riddell's formula for unlabeled
graph (Sloane and Plouffe 1995, p. 20).
There are also important number theoretic applications of the Euler transform. For example, if there are 
kinds of parts of size 1, 
kinds of parts of size 2, etc., in a given type of partition,
then the Euler transform 
of 
is the number of partitions of 
into these integer parts. For example, if 
for all 
, then 
is the number of partitions of 
into integer parts. Similarly, if 
for 
prime and 
for 
composite, then 
is the number of partitions of 
into prime parts (Sloane and Plouffe 1995, p. 21). Other
applications are given by Andrews (1986), Andrews and Baxter (1989), and Cameron
(1989).
