Gauss's Continued Fraction

Author

Eric W. Weisstein
August 27, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/NumberTheory/GausssContinuedFraction.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/GausssContinuedFraction.html.

©2008 Wolfram Research, Inc. except for portions noted otherwise

Continued Fraction

(((Subscript[\[Null], 2])(Subscript[F, 1])(a,b+1;c+1;z))/((Subscript[\[Null], 2])(Subscript[F, 1])(a,b;c;z)))==(1/(1-((((a(c-b))/(c(c+1)))z)/(1-(((((b+1)(c-a+1))/((c+1)(c+2)))z)/(1-(((((a+1)(c-b+1))/((c+2)(c+3)))z)/(1-(((((b+2)(c-a+2))/((c+3)(c+4)))z)/(1-…))))))))))

Table[-z If[Mod[n,2]==1,(b+Ceiling[n/2])(c-a+Ceiling[n/2]),(a+Ceiling[n/2])(c-b+Ceiling[n/2])]/(c+n)/(c+n+1),{n,2}]

expr1=1/(1 - ((a(c - b)/(c(c + 1)))*z)/
    (1 - ((((b + 1)*(c - a + 1))/((c + 1)*
         (c + 2)))*z)/
      (1 - ((((a + 1)*(c - b + 1))/((c + 2)*
           (c + 3)))*z)/1)))

expr2=1/(1-(((a(c-b))/(c(c+1)))z)/(1+ContinuedFractionK[-z If[Mod[n,2]==1,(b+Ceiling[n/2])(c-a+Ceiling[n/2]),(a+Ceiling[n/2])(c-b+Ceiling[n/2])]/(c+n)/(c+n+1),1,{n,2}]))

Simplify[expr1-expr2]

With[{a=1.2,b=2.7,c=3I,z=.5},{1/(1-(((a(c-b))/(c(c+1)))z)/(1+ContinuedFractionK[-z If[Mod[n,2]==1,(b+Ceiling[n/2])(c-a+Ceiling[n/2]),(a+Ceiling[n/2])(c-b+Ceiling[n/2])]/(c+n)/(c+n+1),1,{n,10}])),
((Hypergeometric2F1[a,b+1,c+1,z])/(Hypergeometric2F1[a,b,c,z]))
}]

1/(1-(((a(c-b))/(c(c+1)))z)/(1+ContinuedFractionK[-z If[Mod[n,2]==1,(b+Ceiling[n/2])(c-a+Ceiling[n/2]),(a+Ceiling[n/2])(c-b+Ceiling[n/2])]/(c+n)/(c+n+1),1,{n,∞}]))