Paperfolding, Automata, and Rational Functions - Diagonals and ...

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A Mahler Functional Equation Aside. Is the number 0.11011001110010011101100011001001 . . . transcendental? Now consider subtracting the spaced out sequence from the paperfolding sequence: If we denote the paperfolding sequence by f1, f2, f3, . . . then we have verified experimentally that the formal power series F (X) = P∞ h=1 satisfies the functional equation F(X) − F(X 2 ) = X/(1 − X 4 ) . fhX h Once noticed, we see that this is obvious. Inserting an extra positive fold is to replace F (X) by F(X 2 ) <strong>and</strong> to add X/(1 − X 4 ). However, the infinite paperfolding sequence is invariant under the addition of a positive fold. 6
A Mahler Functional Equation Aside. Is the number 0.11011001110010011101100011001001 . . . transcendental? Now consider subtracting the spaced out sequence from the paperfolding sequence: If we denote the paperfolding sequence by f1, f2, f3, . . . then we have verified experimentally that the formal power series F (X) = P∞ h=1 satisfies the functional equation F(X) − F(X 2 ) = X/(1 − X 4 ) . fhX h Once noticed, we see that this is obvious. Inserting an extra positive fold is to replace F (X) by F(X 2 ) <strong>and</strong> to add X/(1 − X 4 ). However, the infinite paperfolding sequence is invariant under the addition of a positive fold. 6
Page 1: Paperfolding, Automata, and Rationa
Page 5 and 6: Paperfolding, Automata, and Rationa
Page 7 and 8: Paperfolding Take a rectangular she
Page 31 and 32: A Mahler Functional Equation Aside.
Page 35: A Mahler Functional Equation Aside.
Page 47 and 48: Next, if we pair the sequence Regul
Page 49 and 50: Next, if we pair the sequence Regul
Page 51 and 52: The uniform, or regular, 2-substitu
Page 61 and 62: Characteristic Functions I found th
Page 67 and 68: An Algebraic Equation in Characteri
Page 75 and 76: These remarks show that the paperfo
Page 77 and 78: These remarks show that the paperfo
Page 79 and 80: The Thue-Morse Sequence 0 1 10 11 1
Page 81 and 82: The Thue-Morse Sequence 0 1 10 11 1
Page 83 and 84: Euler’s Identity and a Functional
Page 85 and 86: Euler’s Identity and a Functional
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Euler’s Identity and a Functional
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An Algebraic Equation The function
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A Counter-example in Analysis I cla
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The Shapiro Sequence Consider, the
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A Remark on the Shapiro Function Se
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Transcendence of Automatic Numbers
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Complexity of a Sequence Given an i
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Complexity of a Sequence Given an i
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Algebraicity and Automaticity The p
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Comments on the Proof The best proo
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But for expansions over the complex
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Power Series in Several Variables R
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Diagonals and Hadamard Products The
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Diagonals and Hadamard Products The
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A Theorem of Furstenberg It follows
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As said, in characteristic zero, ne
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A Beautiful Transcendence Argument
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A Beautiful Transcendence Argument
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Breaking Up in Characteristic p The
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Algebraic power series in character
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Hence, since Fp is finite, there ar
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Hence, since Fp is finite, there ar
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Suppose the series P aνx ν is gen
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So we have: Theorem. P aνx ν ∈
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The Lifting Theorem Looking careful
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Diagonals of Rational Functions Now
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Whilst (i) and (ii) are reasonably
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References MICHEL DEKKING, MICHEL M
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Paperfolding, Automata, and Rational Functions - Diagonals and ...

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