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Using a technique due to Gauss a lot of special functions can be expressed as continued fractions.

What applications of this are there within mathematics and number theory?

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One application mentioned in that page is: Analytic continuation of $_2 F_1$ type. – user58512 Jan 27 '13 at 21:17

Computationally, CF expansions of hypergeometric functions can be very useful. In particular, one might want to remember that the series for hypergeometric functions can, for certain parameter values, be only convergent for arguments within the unit disk. In contrast, the CF expansions usually have a larger domain of convergence, usually a half-plane or a slit plane.

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