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

In: Number Theory in Memory of Eduard Wirsing

Author

Listed:
  • Carsten Elsner

    (FHDW University of Applied Sciences, Institute of Computer Sciences)

  • Christopher Robin Havens

    (PMP Prison Mathematics Project)

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.

Suggested Citation

  • Carsten Elsner & Christopher Robin Havens, 2023. "Diophantine Analysis Around [ 1 , 2 , 3 , … ] $$[1,2,3,\dots ]$$," Springer Books, in: Helmut Maier & JΓΆrn Steuding & Rasa Steuding (ed.), Number Theory in Memory of Eduard Wirsing, pages 119-143, Springer.
    • Handle: RePEc:spr:sprchp:978-3-031-31617-3_9
    DOI: 10.1007/978-3-031-31617-3_9
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