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Posts Tagged ‘continued fractions’

Previously we described all finite-dimensional random algebras with faithful states. In this post we will describe states on the infinite-dimensional ^{\dagger}-algebra \mathbb{C}[x]. Along the way we will run into and connect some beautiful and classical mathematical objects.

A special case of part of the following discussion can be found in an old post on the Catalan numbers.

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Some countability proofs

Theorem: The set of finite sequences of elements of a countable set S is countable.

I like this result because it specializes to several other basic countability results: for example, it implies that countable unions, finite products, and the set of finite subsets of countable sets are countable. I know several proofs of this result and I am honestly curious which ones people prefer.

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I’ve been inspired by The Unapologetic Mathematician (and his pages and pages of archives!) to post more often, at least for the remainder of the summer. So here is a circle of ideas I’ve been playing with for some time.

Let C(x) = \sum_{n \ge 0} C_n x^n be the ordinary generating function for the ordered rooted trees on n+1 vertices (essentially we ignore the root as a vertex). This is one of the familiar definitions of the Catalan numbers. From a species perspective, ordered rooted trees are defined by the functional equation

\displaystyle C(x) = \frac{1}{1 - xC(x)}.

The generating function \frac{1}{1 - x} = 1 + x^2 + ... describes the species \textsc{Seq} of sequences. So what this definition means is that, after tossing out the root, an ordered rooted tree is equivalent to a sequence of ordered rooted trees (counting their roots) in the obvious way; the roots of these trees are precisely the neighbors of the original root. Multiplying out gets us a quadratic equation we can use to find the usual closed form of C(x), but we can instead recursively apply the above to obtain the beautiful continued fraction

\displaystyle C(x) = \frac{1}{1 - \frac{x}{1 - \frac{x}{1 - ...}}}.

Today’s discussion will center around this identity and some of its consequences.

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