Confluent Hypergeometric Function - Application To Continued Fractions

Application To Continued Fractions

By applying a limiting argument to Gauss's continued fraction it can be shown that

\frac{M(a+1,b+1,z)}{M(a,b,z)} = \cfrac{1}{1 - \cfrac{{\displaystyle\frac{b-a}{b(b+1)}z}} {1 + \cfrac{{\displaystyle\frac{a+1}{(b+1)(b+2)}z}} {1 - \cfrac{{\displaystyle\frac{b-a+1}{(b+2)(b+3)}z}} {1 + \cfrac{{\displaystyle\frac{a+2}{(b+3)(b+4)}z}}{1 - \ddots}}}}}

and that this continued fraction converges uniformly to a meromorphic function of z in every bounded domain that does not include a pole.

Read more about this topic:  Confluent Hypergeometric Function

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