Abstract
We show how functions F(z) which satisfy an identity of the form F(α z) = g(F(z)) for some complex number α and some function g(z) give rise to infinite product formulas that generalize Viète's product formula for π. Specifically, using elliptic and trigonometric functions we derive closed form expressions for some of these infinite products. By evaluating the expressions at certain points we obtain formulas expressing infinite products involving nested radicals in terms of well-known constants. In particular, simple infinite products for π and the lemniscate constant are obtained.
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2000 Mathematics Subject Classification: Primary—40A20; Secondary—33E05
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Levin, A. A New Class of Infinite Products Generalizing Viète's Product Formula for π. Ramanujan J 10, 305–324 (2005). https://doi.org/10.1007/s11139-005-4852-z
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DOI: https://doi.org/10.1007/s11139-005-4852-z