Infinite Products Involving Binary Digit Sums

Abstract

Let (u n ) n≥0 denote the Thue-Morse sequence with values ±1. The Woods-Robbins identity below and several of its generalisations are well-known in the literature ∞ ∏ n=0 arXiv:1709.04104v2 [math.NT]

Infinite products related to generalized Thue-Morse sequences

“…As the first main result in this paper, the following theorem generalizes [2, Theorem 2.2 and Corollary 2.3 (i)] (see also [13,Lemma 2] and the equalities ( 6) and (7) in [13,Section 4]).…”

“…Note that for any integer q ≥ 2, the ( q 0, • • • , 0)-Thue-Morse sequence is the trivial 0 ∞ . For q = 2, the only nontrivial case, related to the (0, 1)-Thue-Morse sequence, is already studied in [13] and [2, Section 2]. In the following three examples, we study nontrivial cases for q = 3 in detail, related to the (0, 0, 1), (0, 1, 1) and (0, 1, 0)-Thue-Morse sequences.…”

“…,8 ,11 and16 follow respectively from dividing 5 by 1 , multiplying 7 by 1 , dividing 9 by 6 and dividing 15 by13 14. and 15 follow from taking q = 5, and then a = 1 2 and 3 respectively in (2).…”

More infinite products: Thue–Morse and the gamma function

“…This paper is an extended version of [11]. While we were preparing this extended version, we found the paper [15] which has interesting results on finite (and infinite) sums involving the sum of digits of integers in integer bases.…”