TOPICS
Search

Kummer's Theorem

The identity

_2F_1(x,-x;x+n+1;-1)=(Gamma(x+n+1)Gamma(1/2n+1))/(Gamma(x+1/2n+1)Gamma(n+1)),

or equivalently

_2F_1(alpha,beta;1+alpha-beta;-1)=(Gamma(1+alpha-beta)Gamma(1+1/2alpha))/(Gamma(1+alpha)Gamma(1+1/2alpha-beta)),

where _2F_1(a,b;c;z) is a hypergeometric function and Gamma(z) is the gamma function. This formula was first stated by Kummer (1836, p. 53).

See also

Saalschütz's Theorem

Explore with Wolfram|Alpha

References

Bailey, W. N. "Kummer's Theorem." §2.3 in Generalised Hypergeometric Series. Cambridge, England: Cambridge University Press, pp. 9-10, 1935.Kummer, E. E. "Ueber die hypergeometrische Reihe." J. für Math. 15, 39-83, 1836.

Referenced on Wolfram|Alpha

Kummer's Theorem

Cite this as:

Weisstein, Eric W. "Kummer's Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/KummersTheorem.html

Subject classifications