Consider the following identity which arose while solving a system of integral equations $$ \sum_{i=0}^\infty \frac{2\beta_{2i} f_- (r) + (\alpha_{2i}-\beta_{2i}) f_+ (r)}{2i+1} = 1 \, , \tag{1} $$ wherein $$ f_\pm (r) = {}_3F_2 \left( \pm \frac{1}{2}, \frac{1}{2}, -\frac{1}{2}-i ; 1 ,\frac{1}{2} -i ; r^2\right) $$ is expressed in terms of the hypergeometric function. Here, $\alpha_{2i}$ and $\beta_{2i}$, $i = 0, 1, \dots$, are unknown series coefficients.
In addition, $r \in [0,1]$.
For the determination of $\alpha_{2i}$ and $\beta_{2i}$, $i = 0, 1, \dots, N$, I tried to solve the resulting linear system of $2(N+1)$ for the unknown coefficients, after expanding the left-hand side of Eq. (1) up to $\mathcal{O} \left( r^{4(N+1)} \right)$ and identify term by term in powers of $r$.
However, it appears that the rank of the resulting matrix is $N+2$ and not $2(N+1)$, making is impossible to solve for both $\alpha_{2i}$ and $\beta_{2i}$. Accordingly, a solution of the linear system of equations is only possible if a relationship between $\alpha_{2i}$ and $\beta_{2i}$ is known.
Any hint that could help solve this problem would be highly appreciated.
Thank you!