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Gauss's Continued Fraction

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Gauss's continued fraction is given by the continued fraction

(_2F_1(a,b+1;c+1;z))/(_2F_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-...))))),

where _2F_1(a,b;c;z) is a hypergeometric function. Many analytic expressions for continued fractions of functions can be derived from this formula.

See also

Continued Fraction

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References

Borwein, J.; Bailey, D.; and Girgensohn, R. "Gauss's Continued Fraction." §1.8.3 in Experimentation in Mathematics: Computational Paths to Discovery. Wellesley, MA: A K Peters, pp. 31-34, 2004.Wall, H. S. "The Continued Fraction of Gauss." Ch. 18 in Analytic Theory of Continued Fractions. New York: Chelsea, pp. 335-361, 1948.

Referenced on Wolfram|Alpha

Gauss's Continued Fraction

Cite this as:

Weisstein, Eric W. "Gauss's Continued Fraction." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/GausssContinuedFraction.html

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