Diophantine Analysis Around [ 1, 2, 3, … ] $$[1,2,3,\dots ]$$

Carsten Elsner () and Christopher Robin Havens ()
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Carsten Elsner: FHDW University of Applied Sciences, Institute of Computer Sciences
Christopher Robin Havens: PMP Prison Mathematics Project

A chapter in Number Theory in Memory of Eduard Wirsing, 2023, pp 119-143 from Springer

Abstract: Abstract The transcendence of the regular infinite continued fraction 𝔷 : = [ 1 , 2 , 3 , 4 , 5 , … ] $$\text{ {$\mathfrak {z}$}} := [1,2,3,4,5,\dots ]$$ was first proven by C. L. Siegel in 1929. The value of 𝔷 $$\text{ {$\mathfrak {z}$}}$$ is a ratio of the values of modified Bessel functions. In this paper our diophantine analysis around 𝔷 $$\text{ {$\mathfrak {z}$}}$$ takes its starting point with its rational convergents and deals with an asymptotic approximation formula for 𝔷 $$\text{ {$\mathfrak {z}$}}$$ and with the construction of a sequence of quadratically irrational approximations using these convergents. Finally, we study various error sums for 𝔷 $$\text{ {$\mathfrak {z}$}}$$ which are also defined by the rational convergents.

Keywords: Continued fractions; Error sums; Recurrences; Bessel functions (search for similar items in EconPapers)
Date: 2023
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-031-31617-3_9

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DOI: 10.1007/978-3-031-31617-3_9

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