A New Class of Infinite Products Generalizing Viète's Product Formula for π
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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.
Key Wordsinfinite products nested radicals elliptic functions
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- 2.P. Beckmann, A History of Pi, 5th edn. The Golem Press, Boulder Colorado, 1982.Google Scholar
- 5.B.C. Berndt and R.A. Rankin (eds.), Ramanujan: Essays and Surveys, American Mathematical Society, 2001.Google Scholar
- 6.J. Borwein and P. Borwein, Pi and the AGM. A Study in Analytic Number Theory and Computational Complexity, Canadian Mathematical Society Series of Monographs and Advanced Texts 4,John Wiley and Sons, Inc., New York, 1998.Google Scholar
- 11.T.J. Osler, “Interesting finite and infinite products fromsimple algebraic identities,” Math. Gaz. (to appear).Google Scholar
- 13.J.H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, Graduate Texts in Mathematics 151, Springer-Verlag, New York, 1994.Google Scholar
- 15.J. Todd, “The lemniscate constants,” Comm. ACM 18 (1975), 14–19; corrigendum, ibid. 18(8), 462.Google Scholar