Consider the equation
$(2+z^2) f(z) + 3 z + 4 = f(z+1)$
Such that $f(z)$ is analytic near the positive real axis and the functional equation holds for real $z>0$.
Can we express Some solutions for $f(z)$ ?
Notice $g(z+1) = (z^2) g(z)$ is expressed by the square of the gamma function.