2014/06/28 by Jean‐Paul Allouche, Allouche, Jean-Paul
Computer Science · #Algorithms and Data Compression #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT) #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.1406.7407
openalex publication_date 2014/06/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Taking the product of (2n+1)/(2n+2) raised to the power +1 or -1 according to the n-th term of the Thue-Morse sequence gives rise to an infinite product P while replacing (2n+1)/(2n+2) with (2n)/(2n+1) yields an infinite product Q, where P = (1/2)(4/3)(6/5)(7/8)(10/9)... and Q = (2/3)(5/4)(7/6)(8/9)(11/10)... Though it is known that P = 2-1/2, nothing is known about Q. Looking at the corresponding question when the Thue-Morse sequence is replaced by the regular paperfolding sequence, we obtain two infinite products A and B, where A = (1/2)(3/4)(6/5)(7/8)(9/10)... and B = (2/3)(4/5)(7/6)(8/9)(10/11)... Here nothing is known for A, but we give a closed form for B that involves the value of the gamma function at 1/4. We then prove general results where (2n+1)/(2n+2) or (2n)/(2n+1) are replaced by specific rational functions. The corresponding infinite products have a closed form involving gamma values. In some cases there is no explicit gamma value occurring in the closed-form formula, but only trigonometric functions.