Is there any way to simplify ${_2F_1}(a, 1, a+2k, -1)$?

Question

Is there any way to simplify or express differently

$${_2F_1}(a,1,a+2k,-1)$$?

where ${_2F_1}$ is the hypergeometric function ?

I am trying to simplify a very messy expression so any kind of different representation could be a nice first step.

Answer 1

HINT:

$${_2F_1}(a,1,a+2k,-1)=\frac{\Gamma(0)\left(\Gamma(a+2k)\text{P}^{-1+a+2k,1-2k}_{-1}(3)\right)}{\Gamma(-1+a+2k)}$$

With $P_n^{(a,b)}(x)$ is the Jacobi polynomial!