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 {k^n}" src="https://mathworld.wolfram.com/images/equations/JacksonsDifferenceFan/Inline2.gif" style="height:15px; width:23px" /> 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 {a_i}_(i=0)^n" src="https://mathworld.wolfram.com/images/equations/JacksonsDifferenceFan/Inline9.gif" style="height:17px; width:38px" />, 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 {b_i}_(i=0)^n" src="https://mathworld.wolfram.com/images/equations/JacksonsDifferenceFan/Inline15.gif" style="height:17px; width:38px" /> is given by

When , this gives the inverse binomial transform of {b_i}_(i=0)^n" src="https://mathworld.wolfram.com/images/equations/JacksonsDifferenceFan/Inline17.gif" style="height:17px; width:38px" />.

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.