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Pentagonal Number Theorem

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The pentagonal number theorem states that for |x|<1,

product_(k=1)^(infty)(1-x^k)=sum_(k=-infty)^(infty)(-1)^kx^(k(3k+1)/2)
(1)
=1+sum_(k=1)^(infty)(-1)^k[x^(k(3k-1)/2)+x^(k(3k+1)/2)]
(2)
=(x)_infty
(3)
=1-x-x^2+x^5+x^7-x^(12)-x^(15)+x^(22)+x^(26)-...
(4)

(OEIS A010815), where 0, 1, 2, 5, 7, 12, 15, 22, 26, ... (OEIS A001318) are generalized pentagonal numbers and (x)_infty 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, also for |x|<1, are

product_(k=1)^(infty)(1-x^kt)=sum_(n=0)^(infty)((-1)^nx^(n(n+1)/2))/(product_(k=1)^(n)(1-x^k))t^n
(5)
=((t;x)_infty)/(1-t)
(6)
product_(k=1)^(infty)(1-x^kt)^(-1)=sum_(n=0)^(infty)(x^n)/(product_(k=1)^(n)(1-x^k))t^n
(7)
=(1-t)/((t;x)_infty),
(8)

where the first identity holds for every complex number t, and the second holds for |xt|<1. The quotients at t=1 are interpreted by continuity, and the empty product for n=0 is 1.

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, 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.Bailey, W. N. Generalised Hypergeometric Series. Cambridge, England: Cambridge University Press, p. 72, 1935.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 (1-x)(1-xx)(1-x^3)(1-x^4)(1-x^5) etc. in seriem simplicem." Acta Academiae Scientarum Imperialis Petropolitinae 1780, pp. 47-55, 1783. Opera Omnia, Series Prima, Vol. 3. pp. 472-479. English translation: "The Expansion of the Infinite Product (1-x)(1-xx)(1-x^3)(1-x^4)(1-x^5)(1-x^6) etc. into a Single Series" (Trans. J. Bell). 4 Dec 2004. https://arxiv.org/abs/math/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."

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Pentagonal Number Theorem

Cite this as:

Weisstein, Eric W. "Pentagonal Number Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PentagonalNumberTheorem.html

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