List of hypergeometric identities

From Wikipedia, the free encyclopedia

In mathematics, the hypergeometric series is involved in a number of classical hypergeometric identities. This article provides an incomplete listing of some of these, referred to as Bailey's list, in reference to W. N. Bailey.

The contemporary method to verify new identities on hypergeometric functions is computational and has been automated.

Contents

[edit] Dougall's formula

Dougall's formula is

\sum_{n=-\infty}^\infty \frac {\Gamma(a+n) \Gamma(b+n)}{\Gamma(c+n)\Gamma(d+n)} = \frac {\pi^2}{\sin (\pi a) \sin (\pi b)} \frac {\Gamma (c+d-a-b-1)}{\Gamma(c-a) \Gamma(d-a) \Gamma(c-b) \Gamma(d-b)}

[edit] Gauss's theorem

Gauss's theorem, named for Carl Friedrich Gauss, is the identity

\;_2F_1 (a,b;c;1)= \frac{\Gamma(c)\Gamma(c-a-b)}{\Gamma(c-a)\Gamma(c-b)}

[edit] Euler's transformation

Euler's transformation is

(1-z)^{a+b-c} \;_2F_1 (a,b;c;z) = \;_2F_1 (c-a, c-b;c ; z)

[edit] Kummer's theorem

Kummer's theorem, named for Ernst Kummer, is

\;_2F_1 (a,b;1+a-b;-1)= \frac{\Gamma(1+a-b)\Gamma(1+a/2)}{\Gamma(1+a)\Gamma(1-b+a/2)}

which follows from

\;_2F_1 (a,b;1+a-b;z)= (1-z)^{-a} \;_2F_1 \left( \frac{a}{2}, \frac{1+a}{2} -b; 1+a-b; \frac{-4z}{(1-z)^2} \right)

[edit] Gauss's second summation theorem

Gauss's second summation theorem is

\;_2F_1 (a,b;1/2+a/2+b/2;1/2)= \frac{\Gamma(1/2)\Gamma(1/2+a/2+b/2)}{\Gamma(1/2+a/2)\Gamma(1+b/2)}


[edit] References

  • W. N. Bailey, Generalized Hypergeometric Series, (1935) Cambridge Tracts in Mathematics and Mathematical Physics, No.32, Cambridge University Press, Cambridge.