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.
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.