Purely periodic continued fraction

A continued fraction is purely periodic when its repeating block begins at its first partial quotient:

[\overline{a_{0}, \dots, a_{n}}] = [a_{0}, \dots, a_{n}, a_{0}, \dots, a_{n}, \dots] .

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- Reversed-period identity for a purely periodic continued fraction Purely periodic continued fraction

Reversed-period identity for a purely periodic continued fraction

Let

θ = [\overline{a_{0}, \dots, a_{n}}]

, and let

θ^{'}

be its algebraic conjugate, the other fixed point of the Möbius transformation represented by

M (a_{0}) \dots M (a_{n})

. Then

- \frac{1}{θ ^{'}} = [\overline{a_{n}, \dots, a_{0}}] .

The identity follows by transposing the continued-fraction matrix and substituting

X = - 1/ Y

in its fixed-point equation.

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