The shuffle product of two words is the formal sum of all interleavings of their letters that preserve the order within each word,
counted with multiplicity. Writing this product as 
, it satisfies
 |
(1)
|
where 
and 
are letters of an alphabet, 
and 
are words, and juxtaposition denotes
concatenation. The word 
of length zero satisfies 
. For example,
 |
(2)
|
Repeated letters can give repeated summands, as in 
. There are 
interleavings of words of
lengths 
and 
,
where the count includes multiplicities. Extending the product bilinearly to formal
linear combinations of words
over a commutative ring gives an associative commutative algebra, called the shuffle algebra
(Luque and Thibon 2002).
A realization assigns to a word 
the multiple integral
 |
(3)
|
where the 
are Lebesgue integrable on 
and 
. Dividing the Cartesian
product of two integration regions according to the relative order of their variables
gives 
,
with 
extended linearly (Chen 1954, Luque and Thibon 2002).
For example, taking 
for positive real numbers 
gives the rational identity
 |
(4)
|
where 
is the symmetric group (Luque and Thibon 2002,
Trott 2006). Each summand is the integral over one ordering
of the variables in the unit hypercube, while the full
integral factors as 
. The identity also holds for complex numbers 
whenever all nonempty subset sums are nonzero, since it is an
identity of rational functions.
