Jackson's Difference Fan

If, after constructing a difference table, no clear pattern emerges, turn the paper through an angle of 60 degrees and compute a new table. If necessary, repeat the process. Each rotation reduces powers by 1, so the sequence {k^n} multiplied by any polynomial in n is reduced to 0s by a k-fold difference fan.

Call Jackson's difference fan sequence transform the J-transform, and define J^k(a)_n as the k-th J-transform of the sequence {a_i}_(i=0)^n, where a and k are complex numbers. This is denoted

 J^k(a)_n=sum_(i=0)^n(-k)^(n-i)(n; i)a_i=b_n.

When k=1, this is known as the binomial transform of the sequence. Greater values of k give greater depths of this fanning process.

The inverse J-transform of the sequence {b_i}_(i=0)^n is given by

 J^(-k)(b)_n=sum_(i=0)^nk^(n-i)(n; i)b_i=a_n.

When k=1, this gives the inverse binomial transform of {b_i}_(i=0)^n.

See also

Difference Table

Portions of this entry contributed by Jason Shields

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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.

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