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Euler Identity

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For |z|<1,

product_(k=1)^infty(1+z^k)=product_(k=1)^infty(1-z^(2k-1))^(-1).
(1)

Both of these have closed form representation

1/2(-1;z)_infty,
(2)

where (a;q)_infty is a q-Pochhammer symbol.

Expanding and taking a series expansion about zero for either side gives

1+z+z^2+2z^3+2z^4+3z^5+4z^6+5z^7+...,
(3)

giving 1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18, 22, 27, ... (OEIS A000009), i.e., the number of partitions of n into distinct parts Q(n).

See also

Euler Formula, Jacobi Triple Product, Partition Function Q, q-Series

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References

Bailey, W. N. Generalised Hypergeometric Series. Cambridge, England: Cambridge University Press, p. 72, 1935.Franklin, F. "Sur le developpement du produit infini (1-x)(1-x^2)(1-x^3)(1-x^4)...." Comptes Rendus Acad. Sci. Paris 92, 448-450, 1881.Hardy, G. H. §6.2 in Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, pp. 83-85, 1999.Hardy, G. H. and Wright, E. M. §19.11 in An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, 1979.MacMahon, P. A. Combinatory Analysis, Vol. 2. New York: Chelsea, pp. 21-23, 1960.Nagell, T. Introduction to Number Theory. New York: Wiley, p. 55, 1951.Sloane, N. J. A. Sequence A000009/M0281 in "The On-Line Encyclopedia of Integer Sequences."

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Euler Identity

Cite this as:

Weisstein, Eric W. "Euler Identity." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/EulerIdentity.html

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