Gauss's Continued Fraction - Derivation

Derivation

Let be a sequence of analytic functions so that

for all, where each is a constant.

Then

.

Setting ,

,

So

.

Repeating this ad infinitum produces the continued fraction expression

In Gauss's continued fraction, the functions are hypergeometric functions of the form, and, and the equations arise as identities between functions where the parameters differ by integer amounts. These identities can be proved in several ways, for example by expanding out the series and comparing coefficients, or by taking the derivative in several ways and eliminating it from the equations generated.