Pentagonal Number Theorem

			(1)
			(2)
			(3)
			(4)

(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

			(5)
			(6)
			(7)
			(8)

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. https://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."