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Transcendence of reciprocal sums of binary recurrences

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Abstract

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.

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Authors and Affiliations

  1. Department of Mathematics, Keio University, Hiyoshi 3–14–1, Kohoku-ku, Yokohama, 223-0061, Japan

    Tomoaki Kanoko, Takeshi Kurosawa & Iekata Shiokawa

Authors
  1. Tomoaki Kanoko
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  2. Takeshi Kurosawa
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  3. Iekata Shiokawa
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Correspondence to Takeshi Kurosawa.

Additional information

Communicated by U. Zannier.

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Kanoko, T., Kurosawa, T. & Shiokawa, I. Transcendence of reciprocal sums of binary recurrences. Monatsh Math 157, 323–334 (2009). https://doi.org/10.1007/s00605-008-0073-z

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  • DOI: https://doi.org/10.1007/s00605-008-0073-z

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