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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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A cubic counterpart of Jacobi’s identity and the AGM
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by J. M. Borwein and P. B. Borwein PDF
Trans. Amer. Math. Soc. 323 (1991), 691-701 Request permission

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 \[ {a_{n + 1}}: = \frac {{{a_n} + 2{b_n}}} {3}\quad {\text {and}}\quad {b_{n + 1}}: = \sqrt [3]{{{b_n}\left ( {\frac {{a_n^2 + {a_n}{b_n} + b_n^2}} {3}} \right ).}}\] 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.
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Additional Information
  • © Copyright 1991 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 323 (1991), 691-701
  • MSC: Primary 33C75; Secondary 11F11, 11Y60, 33C05
  • DOI: https://doi.org/10.1090/S0002-9947-1991-1010408-0
  • MathSciNet review: 1010408