The partition-theoretic Euler identity states that, for
(1)
This identity is distinct from the use of "Euler's identity" for the special case Euler formula . Both infinite products
have the closed form representation
(2)
where 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
See also Euler Formula ,
Jacobi Triple Product ,
Partition Function Q ,
q -SeriesExplore with Wolfram|Alpha
References Bailey, W. N. Generalised Hypergeometric Series. Franklin, F. "Sur le developpement du produit infini C. R. 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. Hardy, G. H. and Wright, E. M.
§19.11 in An
Introduction to the Theory of Numbers, 5th ed. MacMahon, P. A. Combinatory
Analysis, Vol. 2. Nagell,
T. Introduction
to Number Theory. Sloane,
N. J. A. Sequence A000009 /M0281
in "The On-Line Encyclopedia of Integer Sequences." Referenced
on Wolfram|Alpha Euler Identity
Cite this as:
Weisstein, Eric W. "Euler Identity." From
MathWorld https://mathworld.wolfram.com/EulerIdentity.html
Subject classifications
,
of the Euler formula. Both infinite products
have the closed form representation
is a q-Pochhammer symbol.
into distinct parts 
.
."
C. R. Acad. Sci. Paris 92, 448-450, 1881.Hardy, G. H.
§6.2 in