Monatshefte für Mathematik

, Volume 157, Issue 4, pp 323–334 | Cite as

Transcendence of reciprocal sums of binary recurrences

  • Tomoaki Kanoko
  • Takeshi Kurosawa
  • Iekata Shiokawa


Let {R n }n≥0 be a binary linear recurrence defined by R n+2 = A R n+1 + B R n (n ≥ 0), where A, B, R 0, R 1 are integers and Δ = A 2 + 4B > 0. We give necessary and sufficient conditions for the transcendence of the numbers
$$\sum_{k\geq 0}{}^{\prime}\frac{a_k}{R_{r^k}+b},$$
where r ≥ 2 is an integer, {a k }k ≥ 0 is a linear recurrence of algebraic numbers, and b is an algebraic number. We remove the condition assumed in the preceding work that A ≠ 0 and Δ is not a perfect square.


Transcendence Binary linear recurrence Mahler function 

Mathematics Subject Classification (2000)

11J81 11J91 


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Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  • Tomoaki Kanoko
    • 1
  • Takeshi Kurosawa
    • 1
  • Iekata Shiokawa
    • 1
  1. 1.Department of MathematicsKeio UniversityKohoku-ku, YokohamaJapan

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