Jackson's Difference Fan

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 .

REFERENCES:

Conway, J. H. and Guy, R. K. "Jackson's Difference Fans." In The Book of Numbers. New York: Springer-Verlag, pp. 84-85, 1996.