@Article{         10.2307/2001551,
  Author        = "Borwein, J. M. and Borwein, P. B.",
  Abstract      = "We produce exact cubic analogues of Jacobi's celebrated theta function identity and of the arithmetic-geometric mean iteration of Gauss and Legendre. The iteration in question is $$a\_{n+1}: = \frac{a\_n + 2b\_n}{3} \text \,{and} b\_{n+1}: = \root 3 \of {b\_n \Bigg(\frac{a^2\_n + a\_nb\_n + b^2\_n}{3}\Bigg)}$$. The limit of this iteration is identified in terms of the hypergeometric function $\_2 F\_1(1/3, 2/3; 1; \cdot)$, which supports a particularly simple cubic transformation.",
  date-added    = "2018-04-30 10:54:17 +0000",
  date-modified = "2018-04-30 10:54:31 +0000",
  ISSN          = "00029947",
  Journal       = "Transactions of the American Mathematical Society",
  Keywords      = "hypergeometric functions",
  Number        = "2",
  Pages         = "691--701",
  Publisher     = "American Mathematical Society",
  Title         = "A Cubic Counterpart of Jacobi's Identity and the AGM",
  URL           = "http://www.jstor.org/stable/2001551",
  Volume        = "323",
  Year          = "1991",
  bdsk-url-1    = "http://www.jstor.org/stable/2001551",
  File          = "cubic agm (1991) (0) - a - a - w.pdf"
}

@Article{ 10.2307/2001551, Author = "Borwein, J. M. and Borwein, P. B.", Abstract = "We produce exact cubic analogues of Jacobi's celebrated theta function identity and of the arithmetic-geometric mean iteration of Gauss and Legendre. The iteration in question is $$a_{n+1}: = \frac{a_n + 2b_n}{3} \text \,{and} b_{n+1}: = \root 3 \of {b_n \Bigg(\frac{a^2_n + a_nb_n + b^2_n}{3}\Bigg)}$$. The limit of this iteration is identified in terms of the hypergeometric function $_2 F_1(1/3, 2/3; 1; \cdot)$, which supports a particularly simple cubic transformation.", date-added = "2018-04-30 10:54:17 +0000", date-modified = "2018-04-30 10:54:31 +0000", ISSN = "00029947", Journal = "Transactions of the American Mathematical Society", Keywords = "hypergeometric functions", Number = "2", Pages = "691--701", Publisher = "American Mathematical Society", Title = "A Cubic Counterpart of Jacobi's Identity and the AGM", URL = "http://www.jstor.org/stable/2001551", Volume = "323", Year = "1991", bdsk-url-1 = "http://www.jstor.org/stable/2001551", File = "cubic agm (1991) (0) - a - a - w.pdf" }