More infinite products: Thue–Morse and the gamma function

Letting ( t n ) denote the Thue–Morse sequence with values 0, 1, we note that the Woods–Robbins product ∏ n ≥ 0 2 n + 1 2 n + 2 ( - 1 ) t n = 2 - 1 / 2 involves a rational function in n and the ± 1 Thue–Morse sequence ( ( - 1 ) t n ) n ≥ 0 . The purpose of this paper is twofold. On the one hand, we...

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Bibliographic Details
Published in:The Ramanujan journal Vol. 49; no. 1; pp. 115 - 128
Main Authors: Allouche, J.-P., Riasat, S., Shallit, J.
Format: Journal Article
Language:English
Published: New York Springer US 01.05.2019
Springer Nature B.V
Springer Verlag
Subjects:
Combinatorics
Constants
Field Theory and Polynomials
Fourier Analysis
Functions of a Complex Variable
Gamma function
Mathematics
Mathematics and Statistics
Number Theory
Rational functions
11B83
Prouhet–Thue–Morse sequence
11B85
11A63
68R15
Woods and Robbins product
Closed formulas for infinite products
ISSN:1382-4090, 1572-9303
Online Access:Get full text
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Summary:Letting ( t n ) denote the Thue–Morse sequence with values 0, 1, we note that the Woods–Robbins product ∏ n ≥ 0 2 n + 1 2 n + 2 ( - 1 ) t n = 2 - 1 / 2 involves a rational function in n and the ± 1 Thue–Morse sequence ( ( - 1 ) t n ) n ≥ 0 . The purpose of this paper is twofold. On the one hand, we try to find other rational functions for which similar infinite products involving the ± 1 Thue–Morse sequence have an expression in terms of known constants. On the other hand, we also try to find (possibly different) rational functions R for which the infinite product ∏ R ( n ) t n also has an expression in terms of known constants.
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ISSN:1382-4090
1572-9303
DOI:10.1007/s11139-017-9981-7