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Algebraic independence of the generating functions of Stern’s sequence and of its twist

Peter Bundschuh, Keijo Väänänen (2013)

Journal de Théorie des Nombres de Bordeaux

Very recently, the generating function A ( z ) of the Stern sequence ( a n ) n ≥ 0 , defined by a 0 : = 0 , a 1 : = 1 , and a 2 n : = a n , a 2 n + 1 : = a n + a n + 1 for any integer n > 0 , has been considered from the arithmetical point of view. Coons [8] proved the transcendence of A ( α ) for every algebraic α with 0 < | α | < 1 , and this result was generalized in [6] to the effect that, for the same α ’s, all numbers A ( α ) , A ′ ( α ) , A ′ ′ ( α ) , ... are algebraically independent. At about the same time, Bacher [4] studied the twisted version ( b n ) of Stern’s sequence, defined by b 0 : = 0 , b 1 : = 1 , and b 2 n : = - b n , b 2 n + 1 : = - ( b n + b n + 1 ) for any n > 0 .The aim of our paper is to show...

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