I need to compute for these special polynomials that are linear combinations of certain Jacobi Polynomials and are associated to some convergence acceleration methods of alternating sequences,
$$P_{2n}(\zeta) = \sum_{k=1}^n(-1)^{k-1}\ C_{n,k}\cdot\ _2F_1(-2k,2k;1;\zeta)$$
where
$$\ C_{n,k} = \sum_{\ell=k}^n \binom{n}{\ell }\binom{2\ell }{\ell -k}\frac{\ell ^{n+1}}{(n+1)!}\cdot\frac{k^{2\ell }}{(2\ell )!}$$
this limit $L(\zeta)=\lim_{n\rightarrow\infty}|P_{2n}(\zeta)|^\frac1{2n}$ for $-1\le\zeta\le 0$ or already better, if it is possible, the leading term of the asymptotic expansion. I am particularly interested in $L(-1)$. I know that this is obtained finding the maximum summand, but this is a bivariate nonlinear optimization problem by setting $k=n\cdot t$ and $\ell=n\cdot u$ with $0\le t\le 1$ and $t\le u\le 1$ and applying Stirling approximation. I have not been able to get the solution already .
Numerical tests up to $n=5000$ give a value $L(-1)\sim12.9$. It is conjectured that it is $W_0(e^{-1})^{-2}=12.89615346...$ where $W_0(x)$ is the main branch of the Lambert $W$ function.
Any suggestions are welcome