Monatshefte für Mathematik

, Volume 157, Issue 4, pp 323–334

Transcendence of reciprocal sums of binary recurrences


  • Tomoaki Kanoko
    • Department of MathematicsKeio University
    • Department of MathematicsKeio University
  • Iekata Shiokawa
    • Department of MathematicsKeio University

DOI: 10.1007/s00605-008-0073-z

Cite this article as:
Kanoko, T., Kurosawa, T. & Shiokawa, I. Monatsh Math (2009) 157: 323. doi:10.1007/s00605-008-0073-z


Let {Rn}n≥0 be a binary linear recurrence defined by Rn+2 = ARn+1 + BRn (n ≥ 0), where A, B, R0, R1 are integers and Δ = A2 + 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, {ak}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.


TranscendenceBinary linear recurrenceMahler function

Mathematics Subject Classification (2000)

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© Springer-Verlag 2008