Eigenfunctions of a Linear Operator I would like to find all functions $f_{\lambda}: \mathbb R \rightarrow \mathbb C$ and scalars $\lambda \in \mathbb C$ that satisfy the following equation:
\begin{equation}
\lambda f_{\lambda}(x) = x (f_{\lambda}(x-1) - f_{\lambda}(x)) + (f_{\lambda}(x+1) - f_{\lambda}(x)), \ \ \ \ \ \forall x \in \mathbb R,
\end{equation}
or equivalently
\begin{equation}
(\lambda + x + 1) f_{\lambda}(x) = x f_{\lambda}(x-1) + f_{\lambda}(x+1), \ \ \ \ \ \forall x \in \mathbb R.
\end{equation}
For example, one solution is $\lambda = 0$ and $f_{\lambda}(x) = 1$ for all $x \in \mathbb R$.
I don't know how to systematically approach this problem.  Any guidance would be appreciated.
 A: My answer is partial, but I’ll try to extend it and it may be useful as initial general look and idea.  
Given $\lambda\in\Bbb C$ the equation defines the function $f_\lambda$ on each coset $[x]=x+\Bbb Z$ of the group $\Bbb R$ with respect to a subgroup $\Bbb Z$. Namely, given $y\in\Bbb R$ for each $n\in\Bbb Z $ put $a_n=f(y+n)$. Then the sequence $\{a_n\}$ satisfies the recurrence 
$$a_{n+1}=(\lambda+y+n+1)a_n-(y+n)a_{n-1},$$ 
which uniquely defines the values of the function $f_\lambda$ on the coset $[y]$ provided we are given values of $f_\lambda(y+m)$ and $f_\lambda(y+m+1)$ for some integer $m$ and the number $y$ is non-integer or $m\le -1$ or we are given also the value of  $f_\lambda(-1)$.
If $\lambda=0$ then the recurrence has a partial solution $a_n=c$. If $\lambda=-1$ then it has a partial solution $a_n=c(y+n-1)$. I guess that if $\lambda$ is a non-positive integer then the recurrence has a solution $a_n=p(n)$, where $p$ is a polynomial of degree $-\lambda$. At least, it can be proved the converse: if the recurrence has a solution $a_n=p(n)$, where $p\not\equiv 0$ is a polynomial then its degree equals $-\lambda$.
We may try to find  other partial solutions and, maybe, even a complete solution. But in order to find it we may need to deal with complex powers like  $n^{-\lambda}$. 
In order to correspond the solutions for different cosets we need to impose additional conditions on the function $f_\lambda$ (that is the domain question, which also imposes restrictions on possible solutions of the recurrence). For instance, we may assume that the function $f_\lambda$ is continuous, rational or polynomial.  
