I have the following recurrence equation:
$$(\mu\ n + \nu) f\_{n} + J\Phi^{\*} \sqrt{n+1}f\_{n+1} + J\Phi\ \sqrt{n}f\_{n-1} = 0$$
for complex numbers $f\_{n}$ where $n = 0,1,2,3,...,\infty$ and complex $\Phi$ and real $\mu, \nu, J$. Is there a way to find a general expression for $f\_{n}$ in terms of $f\_0$?


A few first terms:
$$n = 0:\ \ \ \nu\ f\_{0} + J\Phi^{\*}f\_{1} = 0 \rightarrow f\_{1} = -\frac{\nu}{J\Phi^{\*}}f\_{0}$$
$$n = 1:\ \ \ (\mu + \nu)\ f\_{1} + J\Phi^{\*}\sqrt{2} f\_{2} + J\Phi f\_{0} = 0 \rightarrow f\_{2} = -\frac{\Phi}{\Phi^{\*}\sqrt{2}}f\_{0} - \frac{\mu + \nu}{J\sqrt{2}\Phi^{\*}}f\_{1}$$



find solution of complex number recurrence equation
Multiply the equation with $x^n/sqrt(n!)$. Define a generating function $g(x)$ with coefficients $g\_n:=f\_n/sqrt(n!)$.


You will get a homogeneous differential equation for $g(x)$. 


