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
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.
Similar content being viewed by others
References
Becker P.-G., Töpfer T.: Transcendency results for sums of reciprocals of linear recurrences. Math. Nachr. 168, 5–17 (1994)
Bundschuh P., Pethö A.: Zur Transzendenz gewisser Reihen. Monatsh. Math. 104, 199–223 (1987)
Duverney D.: Irrationality of fast converging series of rational numbers. J. Math. Sci. Tokyo 8, 275–316 (2001)
Duverney D., Kanoko T., Tanaka T.: Transcendence of certain reciprocal sums of linear recurrences. Monatsh. Math. 137, 115–128 (2002)
Duverney D., Shiokawa I.: On series involving Fibonacci and Lucas numbers I. In: Komatsu, T.(eds) Diophantine Analysis and Related Fields. pp. 62–76. Amer Inst of Physics, New York (2008)
Lucas E.: Théorie des fonctions numériques simplement périodiques. Am. J. Math. 1, 184–240 (1878)
Masser D.W.: A vanishing theorem for power series. Invent. Math. 67, 275–296 (1982)
Nishioka K.: Mahler functions and transcendence. Lecture Notes in Mathematics, vol. 1631. Springer, Berlin (1996)
Nishioka K.: Algebraic independence of reciprocal sums of binary recurrences. Monatsh. Math. 123, 135–148 (1997)
Nishioka K.: Algebraic independence of reciprocal sums of binary recurrences II. Monatsh. Math. 136, 123–141 (2002)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by U. Zannier.
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00605-008-0073-z