Generalized Continued Fraction

Notation

The long continued fraction expression displayed in the introduction is probably the most intuitive form for the reader. Unfortunately, it takes up a lot of space in a book (and it's not easy for the typesetter, either). So mathematicians have devised several alternative notations. One convenient way to express a generalized continued fraction looks like this:

$x = b_0+ \frac{a_1}{b_1+}\, \frac{a_2}{b_2+}\, \frac{a_3}{b_3+}\cdots$

Pringsheim wrote a generalized continued fraction this way:

$x = b_0 + \frac{a_1 \mid}{\mid b_1} + \frac{a_2 \mid}{\mid b_2} + \frac{a_3 \mid}{\mid b_3}+\cdots\,$ .

Karl Friedrich Gauss evoked the more familiar infinite product Π when he devised this notation:

$x = b_0 + \underset{i=1}{\overset{\infty}{\mathrm K}} \frac{a_i}{b_i}.\,$

Here the K stands for Kettenbruch, the German word for "continued fraction". This is probably the most compact and convenient way to express continued fractions; however, it is not widely used by English typesetters.

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Generalized Continued Fraction - Notation