Hi everyone,

I think it is the case that

z^{ r + s }/(r+s) F(2r, r+s, 1+r+s, -z) + z^{ s - r} / (r-s) F(2r, r-
s, 1+r-s, -1/z) = B( r-s, r+s)

where F(.,.,.,.) is Gauss's 2F1 hypergeometric function, B(.,.) is the
beta function, r and s are real (may also work if they're complex; I
haven't checked); and z >0.

Does anyone see how to prove this?

Best,
Ian

Hi Ian.

Your result is well known. It's a special case of the following equation :

(Gamma[a] * Gamma[b] / Gamma[c]) * Hypergeometric2F1[a, b, c, z] =
(Gamma[a] * Gamma[b-a] * (-z)^(-a) / Gamma[c-a]) * Hypergeometric2F1[a,
1+a-c, 1+a-b, 1/z] +
(Gamma[b] * Gamma[a-b]* (-z)^(-b) / Gamma[c-b]) * Hypergeometric2F1[b,
1+b-c, 1+b-a, 1/z]

where |arg(z)| < Pi.

This is a relation between three of the 24 Kummer solutions for the
hypergeometric differential equation. In your case, when a = 2r, b = r+s and
c = 1+r+s , the last hypergeometric function becomes 1, and your result can
be easily deduced.

Hope this helps. Regards.

Fran