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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.
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