  We provide a novel local definition for spectrally flowed vertex operators in
the SL(2,$\mathbb{R}$)-WZW model, generalising the proposal of Maldacena and
Ooguri in [arXiv:hep-th/0111180] for the singly-flowed case to all $\omega >
1$. This allows us to establish the precise connection between the computation
of correlators using the so-called spectral flow operator, and the methods
introduced recently by Dei and Eberhardt in [arXiv:2105.12130] based on local
Ward identities. We show that the auxiliary variable $y$ used in the latter
paper arises naturally from a point-splitting procedure in the space-time
coordinate. The recursion relations satisfied by spectrally flowed correlators,
which take the form of partial differential equations in $y$-space, then
correspond to null-state conditions for generalised spectral flowed operators.
We highlight the role of the SL(2,$\mathbb{R}$) series identifications in this
context, and prove the validity of the conjecture put forward in
[arXiv:2105.12130] for $y$-space structure constants of three-point functions
with arbitrary spectral flow charges.
