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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Vydané v:	The Ramanujan journal Ročník 49; číslo 1; s. 115 - 128
Hlavní autori:	Allouche, J.-P., Riasat, S., Shallit, J.
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	New York Springer US 01.05.2019 Springer Nature B.V Springer Verlag
Predmet:	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
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Abstract	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.
AbstractList	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. Letting (tn) denote the Thue–Morse sequence with values 0, 1, we note that the Woods–Robbins product ∏n≥02n+12n+2(-1)tn=2-1/2involves a rational function in n and the ± 1 Thue–Morse sequence ((-1)tn)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)tn also has an expression in terms of known constants. Letting (t n) denote the Thue-Morse sequence with values 0, 1, we note that the Woods-Robbins product n≥0 2n + 1 2n + 2 (−1) tn = 2 −1/2 involves a rational function in n and the ±1 Thue-Morse sequence ((−1) tn) 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) tn also has an expression in terms of known constants.
Author	Allouche, J.-P. Riasat, S. Shallit, J.
Author_xml	– sequence: 1 givenname: J.-P. surname: Allouche fullname: Allouche, J.-P. email: jean-paul.allouche@imj-prg.fr organization: CNRS, IMJ-PRG, Université P. et M. Curie – sequence: 2 givenname: S. surname: Riasat fullname: Riasat, S. organization: School of Computer Science, University of Waterloo – sequence: 3 givenname: J. surname: Shallit fullname: Shallit, J. organization: School of Computer Science, University of Waterloo
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Snippet	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... Letting (tn) denote the Thue–Morse sequence with values 0, 1, we note that the Woods–Robbins product ∏n≥02n+12n+2(-1)tn=2-1/2involves a rational function in n... Letting (t n) denote the Thue-Morse sequence with values 0, 1, we note that the Woods-Robbins product n≥0 2n + 1 2n + 2 (−1) tn = 2 −1/2 involves a rational...
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SubjectTerms	Combinatorics Constants Field Theory and Polynomials Fourier Analysis Functions of a Complex Variable Gamma function Mathematics Mathematics and Statistics Number Theory Rational functions
Title	More infinite products: Thue–Morse and the gamma function
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