Raising a continued fraction to a power

Question

Is it possible to give a generalized formula for raising an continued fraction to a power? This is assuming that the continued fraction is in the form of $$a_0+\frac{b_1}{a_1+\frac{b_2}{a_2+\ddots}}$$ I'd think first to the binomial expansion series, but even just with squaring, the fraction would divide itself up into trees. Is it even possible with a simple case, like $\phi ^2$, where $$\phi=1+\frac{1}{1+\frac{1}{1+\ddots}}$$ We can see that $\phi$ is the golden ratio $\frac{1+\sqrt{5}}{2}$ by simple numerical manipulation, showing us that $\phi ^2=(3+\sqrt{5})/2$. However, by squaring our continued fraction expansion, that much is certainly unclear. Will you help me?

Answer 1

There has been a fair amount of work on continued fraction arithmetic, depending on what you want there are lots of options. It seems like Bill Gosper's multiplication algorithm is pretty close to what you want. This was discussed in an unpublished paper in 1972. There are some slides by Mark J. Dominus here and this also provides a link to Gosper's version. Liardet and Stambul also discussed questions on continued fraction arithmetic in Algebraic Computations with Continued Fractions in 1996.

This list of work on this topic does not stop here and I can provide more references if required.