In previous papers we introduced a class of polynomials which follow the same recursive formula as the Lucas–Lehmer numbers, studying the distribution of their zeros and remarking that this distribution follows a sequence related to the binary Gray code. It allowed us to give an order for all the zeros of every polynomial Ln. In this paper, the zeros, expressed in terms of nested radicals, are used to obtain two formulas for π: the first can be seen as a generalization of a well known formula related to the smallest positive zero of Ln; the second is an exact formula for π achieved thanks to some identities valid for Ln.

π-Formulas and Gray code / Vellucci, Pierluigi; Bersani, Alberto Maria. - In: RICERCHE DI MATEMATICA. - ISSN 1827-3491. - 68:2(2019), pp. 551-569. [10.1007/s11587-018-0426-4]

π-Formulas and Gray code

Pierluigi Vellucci
Primo
;
Alberto Maria Bersani
Secondo
2019

Abstract

In previous papers we introduced a class of polynomials which follow the same recursive formula as the Lucas–Lehmer numbers, studying the distribution of their zeros and remarking that this distribution follows a sequence related to the binary Gray code. It allowed us to give an order for all the zeros of every polynomial Ln. In this paper, the zeros, expressed in terms of nested radicals, are used to obtain two formulas for π: the first can be seen as a generalization of a well known formula related to the smallest positive zero of Ln; the second is an exact formula for π achieved thanks to some identities valid for Ln.
π-formulas; Gray code; continued roots; nested square roots; zeros of chebyshev polynomials
01 Pubblicazione su rivista::01a Articolo in rivista
π-Formulas and Gray code / Vellucci, Pierluigi; Bersani, Alberto Maria. - In: RICERCHE DI MATEMATICA. - ISSN 1827-3491. - 68:2(2019), pp. 551-569. [10.1007/s11587-018-0426-4]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11573/1178786
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