As the title says, I am looking for good approximate solution of the equation $$\Gamma(x+a)=k\, \Gamma(x+1) \qquad \text{for} \qquad 0 \leq x \leq a$$ for a given real and positive value of $a$ and $k>0$.
This does not make much problems for a numerical point of view if we consider that the problem is to find the zero of function $$f(x)=\log (\Gamma (x+a))-\log (\Gamma (x+1)) -\alpha \qquad\qquad \alpha=\log(k)$$ but the calculations are not cheap in terms of computer resources.
So, what I am looking is a good approximation $x_0$ of the solution such that, using Newton method, $$x_1=x_0-\frac{\log (\Gamma (x_0+a))-\log (\Gamma (x_0+1)) -\alpha }{\psi (x_0+a)-\psi (x_0+1) }$$ be almost the exact solution (say within less than $0.1$% error).
After several attempts, I rescaled the problem using $x=at$ and looked for the zero of function $$g(t)=\frac{\log \left(\frac{\Gamma (a (t+1))}{\Gamma (a) \Gamma (a t+1)}\right)}{\log \left(\frac{\Gamma (2 a)}{\Gamma (a) \Gamma (a+1)}\right)}-\beta$$ where now $0 \leq t \leq 1$ and $0 \leq \beta \leq 1$ (this looks to be more convenient).
Surprising ot not, all the curves are quite similar and "look" more or less like $t^{\gamma(a)}$. For illustration $$\left( \begin{array}{cc} a & \gamma(a) \\ 10 & 0.628739 \\ 20 & 0.608322 \\ 30 & 0.599442 \\ 40 & 0.594307 \\ 50 & 0.590907 \\ 60 & 0.588466 \\ 70 & 0.586617 \\ 80 & 0.585162 \\ 90 & 0.583983 \\ 100 & 0.583006 \end{array} \right)$$
which were obtained minimizing with respect to $\gamma$ $$\Phi_a(\gamma)=\large{\int_0^1} \Bigg[\frac{\log \left(\frac{\Gamma (a (t+1))}{\Gamma (a) \Gamma (a t+1)}\right)}{\log \left(\frac{\Gamma (2 a)}{\Gamma (a) \Gamma (a+1)}\right)}-x^{\gamma}\Bigg]^2\,dt$$
Based on these values, a quick and dirty regression gave $$\gamma(a) \sim \frac{\frac{349}{494}+\frac{90}{1087} a } {1+\frac{101}{701} a}$$ which is more than empirical but which works decently $(R^2>0.999999)$
For illustration, using $a=45.678$ and $\beta=0.7654$, Newton method gives as iterates $$\left( \begin{array}{cc} n & t_n \\ 0 & 0.592046 \\ 1 & 0.611257 \\ 2 & 0.611455 \end{array} \right)$$ wich achieves the goal but at the price of a too high empiricism.
So, my question
Is there any way to better approximate $x_0$ or $\gamma(a)$ based on rigourous mathematics ?