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

Euler’s Identity and a Functional Equation
Fairly obviously, the sequence (sh) is invariant under the uniform
binary substitution θ : 0 ↦→ 01 and 1 ↦→ 10. Now recall Euler’s identity
∞Y “
1 + X 2n”
=
n=0
∞X
h=0
X h = 1
1 − X ,
noting it is just a pleasant way of recalling that the nonegative integers
each have a unique representation in base 2. It will then also be fairly
obvious that
T (X) :=
∞Y “
1 − X 2n”
=
n=0
∞X
(−1) sh h
X ;
and that plainly T (X) = P ∞
h=0 (−1)s hX h satisfies the Mahler functional
equation
(1 − X)T (X 2 ) = T (X) .
h=0
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