$\phi$-Thue-Morse sequences and infinite products

arXiv2006.04909MaRDI QIDQ6342413

Publication date: 8 June 2020

Abstract: In this article we introduce a new approach to compute infinite products defined by automatic sequences involving the Thue-Morse sequence. As examples, for any positive integers

q

and

r

such that

0 l e q r l e q q - 1

, we find infinitely many couples of rational functions

R (x)

and

S (n)

such that

prod_{n=0}^{infty}R(n)^{frac{1+a_n}{2}}S(n)^{frac{1-a_n}{2}}=2cos(frac{2r+1}{2q}pi),

where

(a_{n})_{n i n m a t h b f N}

is the Thue-Morse sequence beginning with

a_{0} = 1, a_{1} = - 1

.

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