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# 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 .

Difference Table

Portions of this entry contributed by Jason Shields

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## 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.

## Referenced on Wolfram|Alpha

Jackson's Difference Fan

## Cite this as:

Shields, Jason and Weisstein, Eric W. "Jackson's Difference Fan." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/JacksonsDifferenceFan.html