If, after constructing a difference table, no clear pattern emerges, turn the paper through an angle
of 
and compute a new table. If necessary, repeat the process. Each rotation
reduces powers by 1, so the sequence 
multiplied by any polynomial
in 
is reduced to 0s by a 
-fold difference fan.
Call Jackson's difference fan sequence transform the 
-transform, and define 
as the 
-th 
-transform of the sequence 
, where 
and 
are complex numbers. This is denoted
 |
When 
,
this is known as the binomial transform of
the sequence. Greater values of 
give greater depths of this fanning process.
The inverse 
-transform
of the sequence 
is given by
 |
When 
,
this gives the inverse binomial transform of
.
