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Gauss's Polynomial Identity

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For even h,

1-(1-x^h)/(1-x)+((1-x^h)(1-x^(h-1)))/((1-x)(1-x^2))-((1-x^h)(1-x^(h-1))(1-x^(h-2)))/((1-x)(1-x^2)(1-x^3))+... =(1-x)(1-x^3)(1-x^5)...(1-x^(h-1))
(1)

(Nagell 1951, p. 176). Writing out symbolically,

sum_(n=0)^h((-1)^nproduct_(k=0)^(n-1)(1-x^(h-k)))/(product_(k=1)^(n)(1-x^k))=product_(k=0)^(h/2-1)1-x^(2k+1),
(2)

which gives

sum_(n=0)^h((-1)^n(x^h;x^(-1))_n)/((x;x)_n)=(x;x^2)_(h/2),
(3)

where (x;a)_n is a q-Pochhammer symbol.

For example, for h=2,

1-(1-x^2)/(1-x)+((1-x)(1-x^2))/((1-x)(1-x^2))=2-(1-x^2)/(1-x)=1-x,
(4)

and for h=4,

1-(1-x^4)/(1-x)+((1-x^4)(1-x^3))/((1-x)(1-x^2))-((1-x^4)(1-x^3)(1-x^2))/((1-x)(1-x^2)(1-x^3))+((1-x)(1-x^2)(1-x^3)(1-x^4))/((1-x)(1-x^2)(1-x^3)(1-x^4)) =2-(2(1-x^4))/(1-x)+((1-x^3)(1-x^4))/((1-x)(1-x^2)) =(1-x)(1-x^3).
(5)

See also

Gauss's Polynomial Theorem, q-Pochhammer Symbol, q-Series

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References

Nagell, T. "A Polynomial Identity of Gauss." §52 in Introduction to Number Theory. New York: Wiley, pp. 174-176, 1951.

Referenced on Wolfram|Alpha

Gauss's Polynomial Identity

Cite this as:

Weisstein, Eric W. "Gauss's Polynomial Identity." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/GausssPolynomialIdentity.html

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