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Gauss-Kummer Series

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A Gauss-Kummer series is defined, for complex numbers h with |h|<=1, by

_2F_1(-1/2,-1/2;1;h^2)=sum_(n=0)^(infty)(1/2; n)^2h^(2n)
(1)
=1+1/4h^2+1/(64)h^4+1/(256)h^6+...
(2)

(OEIS A056981 and A056982), where _2F_1(a,b;c;x) is a hypergeometric function and (n; k) is a binomial coefficient. The series can be derived using Kummer's quadratic transformation. The Gauss-Kummer series is closely related to the perimeter of an ellipse.

See also

Ellipse

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References

Sloane, N. J. A. Sequences A056981 and A056982 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Gauss-Kummer Series

Cite this as:

Weisstein, Eric W. "Gauss-Kummer Series." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Gauss-KummerSeries.html

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