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Transcendence of certain series involving binary linear recurrences

✍ Scribed by Takeshi Kurosawa

Publisher
Elsevier Science
Year
2007
Tongue
English
Weight
228 KB
Volume
123
Category
Article
ISSN
0022-314X

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✦ Synopsis

Duverney and Nishioka [D. Duverney, Ku. Nishioka, An inductive method for proving the transcendence of certain series, Acta Arith. 110 (4) (2003) 305-330] studied the transcendence of k 0

, where E k (z), F k (z) are polynomials, Ξ± is an algebraic number, and r is an integer greater than 1, using an inductive method. We extend their inductive method to the case of several variables. This enables us to prove the transcendence of k 0

, where R n is a binary linear recurrence and {a k } is a sequence of algebraic numbers.

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## Abstract Suppose that {__R~n~__}__n__β‰₯0 is a linear recursive sequence and __d__ β‰₯ 2 is an integer. Under suitable conditions on {__R~n~__} and __d__ we show that and similarly constructed numbers are transcendental. Special attention is given to the case of binary recurrences.