For $w$ a permutation, the associated **(ordinary) Schubert polynomial** $\mathfrak{S}\_w(\textbf{x})$ is a multivariate polynomial that represents for the cohomology class of the Schubert variety $X\_w$ in the flag variety.
There are many references for this fact. See, e.g., this [text](https://books.google.com/books?id=yz7gyKYgIuwC&printsec=frontcover#v=onepage&q&f=false) by Manivel.
One key feature of Schubert polynomials is that they satisfy a recurrence often referred to as the **Lascoux-Schutzenberger transition equation**.
Let $r$ be the largest value such that $w(r)>w(r+1)$ and $s$ be the maximum value so that $w(s) < w(r)$.
We define
$$
I(w,r) = \{i < r: w \lessdot w t\_{ir}\},
$$
where $t\_{rs}$ is the transposition $(r,s)$ and $\lessdot$ indicates a cover relation in Bruhat order.
Then
$$
x\_r \mathfrak{S}\_w(\textbf{x}) = \mathfrak{S}\_{w t\_{rs}}(\textbf{x}) - \sum\_{i \in I(w,r)} \mathfrak{S}\_{w t\_{ir}}(\textbf{x}).
$$
This result dates back to work of Lascoux and Schutzenberger in the 80's and can be derived from Monk's formula.
There is a simple extension allowing one to work with arbitrary descents in $w$.


The **double Schubert polynomial** $\mathfrak{S}(\textbf{x},\textbf{y})$ is a polynomial in two alphabets of variables that represents the equivariant cohomology class of the Schubert variety $X\_w$ in the flag variety.
These polynomials satisfy an analogous recurrence
$$
(x\_r - y\_s) \mathfrak{S}\_w(\textbf{x},\textbf{y}) = \mathfrak{S}\_{w t\_{rs}}(\textbf{x},\textbf{y}) - \sum\_{i \in I(w,r)} \mathfrak{S}\_{w t\_{ir}}(\textbf{x},\textbf{y}).
$$
I've seen this identity proved in a class and can rederive it myself (as well as the more general transition equations), but can't seem to find an explicit reference in the literature. 
I've asked around a bit, but haven't turned anything up.
Can anyone point me to a concrete reference of the transition equations for double Schubert polynomials (or something equivalent) in the literature?


P.S. There is an unpublished preprint by Lascoux called "[Chern and Yang Through Ice](http://www-igm.univ-mlv.fr/~al/ARTICLES/ChernYang.ps.gz)" that describes transition for double Grothendieck polynomials (a stronger result), but I'm wary of giving it as a reference for two reasons.
First, it claims a stronger result that requires a bit of work to translate to my setting.
Second, I find the whole document a great challenge to parse, and am skeptical that a complete proof is presented.



Transition equations for double Schubert polynomials
On pg. 5 of <https://arxiv.org/abs/1909.13777>, Allen Knutson suggests that Lascoux proves a transition formula for double Schubert polynomials in the paper "Transition on Grothendieck polynomials", Physics and Combinatorics 2000, pp. 164-179, available online (if you pay for it) at <https://www.worldscientific.com/doi/abs/10.1142/9789812810007_0007>. But in that sentence maybe Knutson just means, the double version of what Lascoux does.


See also these slides of Alex Yong (which discuss joint work with Knutson which I think Knutson explains in the above arXiv preprint was ultimately unpublished) explaining a Gröbner basis approach: <https://faculty.math.illinois.edu/~ayong/slides_AMS.04.10.18.pdf>.


EDIT:


Okay, so having obtained the Physics and Combinatorics 2000 article "via undisclosed means", in the introduction Lascoux says



> 
> The cohomological computations can easily be extended to two sets of
>  indeterminates, due to the extension of Monk's rule (cf. 10), or translated
>  into planar combinatorial constructions. The same is true for Grothendieck polynomials , but for simplicity, we shall restrict here to only one set of indeterminates.
> 
> 
> 


So I'm not sure how much this is a reference for double anything. The reference 10 for the double Monk's rule is: A. Konhert, S. Veigneau, Using Schubert basis to compute with multivariate polynomials, Adv. Appl. Math. 19, 45-60 (1997). (<https://www.sciencedirect.com/science/article/pii/S0196885897905261>)


EDIT 2:


So doing a bit more googling, I can, oddly enough, find a very specific reference for the transition formula for double Schubert polynomials *of Types B,C,D* (which is of course not what you asked for) in Proposition 6.12 of "Double Schubert polynomials for the classical groups" by Ikeda, Mihalcea, Naruse, Advances in Mathematics 226 (2011) 840–886. (<https://core.ac.uk/download/pdf/81995975.pdf>)


