Following the discussion found here and here, I have been trying to derive the saddlepoint approximation for the generalized chi-square distribution, with the moment generating function defined in Provost's "Quadratic Forms in Random Variables", $$M_Q(t) = \exp\left(t\Sigma_{j=1}^p b_j^2\lambda_j(1-2t\lambda_j)^{-1}\right)\prod_{j=1}^p(1-2t\lambda_j)^{-\frac{1}{2}}$$ for the cases $p=2$ and $p=3$. To try and simplify the process, I tried to derive it for $p=1$, where I managed to solve the saddlepoint equation, but with multiple solutions. I have also tried the $p=2$ case, but got a cubic polynomial that might not be solvable.
Do you have any suggestions for computing a closed-form expression for the saddlepoint approximation for $p=2$ and $p=3$, or would it be better to use computational methods like the one described by J.P. Imhof?
Edit: for clarification, this is what I did for the $p=1$ case. The cumulant generating function in this case is $$ K(t) = t\sum_{j=1}^p \frac{b_j^2\lambda_j}{(1-2t\lambda_j)} + \frac{1}{2}\sum_{j=1}^{p}\ln{1-2t\lambda_j} = \frac{tb_1^2\lambda_1}{1-2t\lambda_1} + \frac{1}{2} \ln{1-2t\lambda_1}, \text{ when } p=1 $$ Then to solve the saddlepoint equation, I set $K'(\hat{s}) = x$ and solve for $\hat{s}$. In this case, this results in the two solutions $$\hat{s}_1 = \frac{0.25 \left(\lambda_1^{2} + 2.0 \lambda_1 x - 2.0 \sqrt{0.25 \lambda_1^{4} + \lambda_1^{3} b_{1}^{2} x}\right)}{\lambda_1^{2} x}, \\ \hat{s}_2 = \frac{0.25 \left(\lambda_1^{2} + 2.0 \lambda_1 x + 2.0 \sqrt{0.25 \lambda_1^{4} + \lambda_1^{3} b_{1}^{2} x}\right)}{\lambda_1^{2} x} $$ I understand that the saddlepoint equation should have a unique solution. In this case, is there a way to discard one of the solutions?
