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Keywords
(4)
Geometric Mean
hypergeometric function
Modular Form
Theta Function
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(2)
A CUBIC COUNTERPART OF JACOBI'S IDENTITY AND THE AGM
Pi and the AGM
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CUBIC MODULAR IDENTITIES OF RAMANUJAN, HYPERGEOMETRIC FUNCTIONS AND ANALOGUES OF THE ARITHMETICGEOMETRIC MEAN ITERATION
CUBIC MODULAR IDENTITIES OF RAMANUJAN, HYPERGEOMETRIC FUNCTIONS AND ANALOGUES OF THE ARITHMETICGEOMETRIC MEAN ITERATION,Frank Garvan
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CUBIC MODULAR IDENTITIES OF RAMANUJAN, HYPERGEOMETRIC FUNCTIONS AND ANALOGUES OF THE ARITHMETICGEOMETRIC MEAN ITERATION
(
Citations: 7
)
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Frank Garvan
There are four values ofs for which the
hypergeometric function
2F1( 1 2 s; 1 2 +s; 1; ) can be parametrized in terms of modular forms; namely, s = 0, 1 3 , 1 4 , 1 6 . For the classical s = 0 case, the parametrization is in terms of the Jacobian theta functions 3(q), 4(q) and is related to the arithmeticgeometric mean iteration of Gauss and Legendre. Analogues of the arithmeticgeometric mean are given for the remaining cases. The case s = 1 6 and its relationship to the work of Ramanu jan is highlighted. The work presented includes various pieces of joint work with combinations of the following: B. Berndt, S. Bhargava, J. Borwein, P. Borwein and M. Hirschhorn.
Published in 1993.
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Annual
Citation Context
(3)
...Elliptic Integral Inequalities, with Applications
G
. D. Anderson, S.L...
...The connection ofK.r/ to the arithmetic‐geometric mean and other mean values was explored in [BB2], [
G
], and [VV]...
...195 196
G
. D. Anderson, S.L. Qiu, and M. K. Vamanamurthy...
...These results are related to normal forms for quartics [Be]. 198
G
. D. Anderson, S.L...
...(2.7) 200
G
. D. Anderson, S.L. Qiu, and M. K. Vamanamurthy...
...‚ ; 202
G
. D. Anderson, S.L. Qiu, and M. K. Vamanamurthy...
...Qiu, and M. K. Vamanamurthy while
G
00 p .r/ may be written as...
...which, by Theorem 1.11(4), is increasing from.0; 1/ onto.0;1/. Hence
G
is increasing and convex...
...Hence G is increasing and convex. 204
G
. D. Anderson, S.L...
...They are grateful to the referee for significant corrections. 206
G
. D. Anderson, S.L...
G. D. Anderson
,
et al.
Elliptic Integral Inequalities, with Applications
...to parametrize a cubic mean iteration whose limit is identied with the the hypergeometric function 2F1( 1 3; 2 3 ; 1; ). For a survey of recent related results see [
9
]...
Frank G. Garvan
.
A Combinatorial Proof of the FarkasKra Theta Function Identities and ...
...For related recent work in this area see Borwein and Borwein (1991), Borwein, Borwein and Garvan (1994), Borwein, Borwein and Garvan (1993), Hirschhorn, Borwein and Garvan (1993), Shen (1994), Shen (manuscript) and
Garvan (1994)
...
...For more discussion see Borwein and Borwein (1987), and
Garvan (1994)
...
FRANK G. GARVANy
.
Ramanujan's theories of elliptic functions to alternative bases  a sy...
References
(8)
The theory of partitions
(
Citations: 1160
)
G. E. Andrews
Published in 1976.
Ramanujan's Theories Of Elliptic Functions To Alternative Bases
(
Citations: 78
)
Bruce C. Berndt
,
S. Bhargava
,
Frank G. Garvan
Published in 1995.
Pi and the AGMA Study in Analytic Number Theory and Computational Complexity
(
Citations: 176
)
J. M. Borwein
,
P. B. Borwein
Published in 1987.
Some Cubic Modular Identities Of Ramanujan
(
Citations: 44
)
J. M. Borwein
,
P. B. Borwein
Published in 1994.
Cranks and tcores
(
Citations: 84
)
Frank. Garvan
,
Dongsu Kim
,
Dennis Stanton
Published in 1990.
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Citations
(7)
Geometric properties of quasiconformal maps
(
Citations: 1
)
Matti Vuorinen
Published in 2007.
Towers of function fields and iterated means
(
Citations: 5
)
Patrick Solé
Journal:
IEEE Transactions on Information Theory  TIT
, vol. 46, no. 4, pp. 15321535, 2000
Elliptic Integral Inequalities, with Applications
(
Citations: 8
)
G. D. Anderson
,
S.L. Qiu
,
M. K. Vamanamurthy
Journal:
Constructive Approximation  CONSTR APPROX
, vol. 14, no. 2, pp. 195207, 1998
A qproduct tutorial for a qseries MAPLE package
(
Citations: 4
)
Frank Garvan
Journal:
World Wide Web  WWW
, 1998
A Combinatorial Proof of the FarkasKra Theta Function Identities and Their Generalizations
(
Citations: 3
)
Frank G. Garvan
Journal:
Journal of Mathematical Analysis and Applications  J MATH ANAL APPL
, vol. 195, no. 2, pp. 354375, 1995