For ,
|
(1) |
Both of these have closed form representation
|
(2) |
where is a q-Pochhammer symbol.
Expanding and taking a series expansion about zero for either side gives
|
(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 into distinct parts .
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 ." 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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Cite this as:
Weisstein, Eric W. "Euler Identity." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/EulerIdentity.html