Analytical solution to equation $ \arctan(x)-k \arctan(x/k)=c$

Question

For the equation:$$\arctan(x)-k \arctan(x/k)=c$$ which is part of a gasdynamics function called Prandtl–Meyer function, it is not difficult to find the solution numerically, however, I'm wondering, does the analytical solution exists? Thanks.
An example for the coefficient is $k=2.4$, $c=-1.4$, the solution is $x=5.57$.

Answer 1

After substituting $a\to i a,b\to ib,c\to e^{-2i c}$ and simplifying the Lagrange inversion:

$$e^{2a\tanh^{-1}(bx)-2\tanh^{-1}(x)}=c\implies x=1+\sum_{n=1}^\infty \frac{(-c)^n}{n!}\left.\frac{d^{n-1}}{dw^{n-1}}\left(\left((w+1)(1-bw)^a (bw+1)^{-a}\right)^n\right)\right|_{w=1}$$

shown here. Now to use the generalized product rule and expanding. The result will be a double hypergeometric series converging for values similar to the “shown here” link. More will develop in the series expansion.