I'm a bit confused by an identity I found in Gradsteyn & Rhyzik. It's entry 8.936.1 (I also consulted the original source which states the same identity) relating Gegenbauer polynomials to Legendre functions.
I don't quite understand why this identity is correct. I obtained
$$ C_\nu ^\alpha(z) = \frac{\Gamma(2\alpha+\nu)\Gamma(\alpha+\frac{1}{2})}{\Gamma(\nu+1)\Gamma(2\alpha)}\bigg(\frac{1}{4}(1-z^2) \bigg)^{1/4-\alpha/2}P^{1/2-\alpha}_{\alpha+\nu-1/2}(z) \qquad (*) $$ Note that the factor in large parentheses is opposite of that of the source, which reads $$ C_\nu ^\alpha(z) = \frac{\Gamma(2\alpha+\nu)\Gamma(\alpha+\frac{1}{2})}{\Gamma(\nu+1)\Gamma(2\alpha)}\bigg(\frac{1}{4}(z^2-1) \bigg)^{1/4-\alpha/2}P^{1/2-\alpha}_{\alpha+\nu-1/2}(z) \qquad (**) $$ I got to $(*)$ this by applying the definition $$P^\alpha_\nu(z) = \frac{1}{\Gamma(1-\alpha)}\bigg( \frac{1+z}{1-z}\bigg)^{\alpha/2}{}_2F_1\left(-\nu,\nu+1;1-\alpha;\frac{1-z}{2}\right); $$ followed by the Euler identity $$ {}_2F_1(a,b;c;u) = (1-u)^{c-a-b}{}_2F_1(c-a,c-b;c;u)$$ to the right hand side. This recovers the definition of $C_\nu ^\alpha$ in terms of the hypergeometric function. Am I missing something, or is there a typo in this source?
Thanks!