Monatshefte für Mathematik

, Volume 157, Issue 4, pp 323–334

Transcendence of reciprocal sums of binary recurrences

  • Tomoaki Kanoko
  • Takeshi Kurosawa
  • Iekata Shiokawa
Article

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

Abstract

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.

Keywords

TranscendenceBinary linear recurrenceMahler function

Mathematics Subject Classification (2000)

11J8111J91

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