(OEIS A010815), where 0, 1, 2, 5, 7, 12, 15, 22, 26, ... (OEIS A001318) are generalized
pentagonal numbers and is a q-Pochhammer
symbol.
This identity was proved by Euler (1783) in a paper presented to the St. Petersburg Academy on August 14, 1775.
Related equalities are
See also
Partition Function P,
Partition Function Q,
Pentagonal
Number,
q-Pochhammer Symbol,
Ramanujan Theta Functions,
Zagier's
Identity
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References
Bailey, W. N. Generalised Hypergeometric Series. Cambridge, England: Cambridge University Press, p. 72,
1935.Bailey, D. H.; Borwein, J. M.; Calkin, N. J.; Girgensohn,
R.; Luke, D. R.; and Moll, V. H. Experimental
Mathematics in Action. Wellesley, MA: A K Peters, pp. 221-222, 2007.Borwein,
J. M. and Borwein, P. B. Pi
& the AGM: A Study in Analytic Number Theory and Computational Complexity.
New York: Wiley, p. 64, 1987.Euler, L. "Evolutio producti
infiniti
etc. in seriem simplicem." Acta Academiae Scientarum Imperialis Petropolitinae
1780, pp. 47-55, 1783. Opera Omnia, Series Prima, Vol. 3. pp. 472-479.
Translated as Bell, J. "The Expansion of the Infinite Product etc. into a Single Series."
Dec. 4, 2004. http://www.arxiv.org/abs/math.HO/0411454/.Hardy,
G. H. Ramanujan:
Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York:
Chelsea, pp. 83-85, 1999.Sloane, N. J. A. Sequences
A001318/M1336 and A010815
in "The On-Line Encyclopedia of Integer Sequences."Referenced
on Wolfram|Alpha
Pentagonal Number Theorem
Cite this as:
Weisstein, Eric W. "Pentagonal Number Theorem."
From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PentagonalNumberTheorem.html
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