Correct notation for cross-level interaction I am wondering if I am using the correct notation for a multilevel model with a cross-level interaction (i.e., a multilevel model with an interaction between level 1 and 2 covariates).
Let $Y_{tj}$ be a test score of the $jth$ student at time $t$. For simplicity, let $gender$ be the only level two (i.e., student level) covariate, and let $months$ be the only level 1 (i.e., repeated observation level) covariate.
Typically in the context of cross-level interactions, which in this example is an interaction between $gender$ and $months$, the interaction is a random effect and can be denoted using the notation below.
(Level-1 model)
$Y_{tj} = \beta_{0j} + \beta_{1j}months + r_{tj}$, where $r_{tj} \tilde{} N(0,\sigma^2)$
(Level-2 model)
$\beta_{0j} = \gamma_{00} + \gamma_{01}gender + u_{0j}$, where $u_{0j} \tilde{} N(0,\tau_{00})$
$\beta_{1j} = \gamma_{10} + \gamma_{11}gender + u_{1j}$,  where $u_{1j} \tilde{} N(0,\tau_{10})$
I am not interested in estimating $\tau_{10}$. Or, put differently, I want the coefficient corresponding to $months$ to be fixed. Given this, is the below notation correct?
$Y_{tj} = \beta_{0j} + \beta_{1}months + r_{tj}$, where $r_{tj} \tilde{} N(0,\sigma^2)$
(Level-2 model)
$\beta_{0j} = \gamma_{00} + \gamma_{01}gender + u_{0j}$, where $u_{0j} \tilde{} N(0,\tau_{00})$
$\beta_{1} = \gamma_{10} + \gamma_{11}gender$
Any comments on my notation are welcome! Though I would also appreciate references to articles estimating such cross-level interactions.

As @Frank notes in a comment, what you show in your question does not evaluate  an interaction between months and gender. If we follow the Wikipedia naming of levels, Level 1 represents the observations at the individual level.* All "fixed-effect" predictors would be included at that level. Your Level-1 model thus should include both of those predictors individually, plus an interaction term between them if that's what you want.
Level 2 then allows for the intercepts and regression coefficients at Level 1 to differ depending on which group or individual is being observed. If you don't want to allow the coefficient(s) for months to differ among individuals, don't include a corresponding random effect.

*When you call the student-level observations "Level 2" you seem to be defining the levels differently from how Wikipedia does. In Wikipedia, Level 1 represents the individual-level observations. I find the numbering of levels to be more confusing than helpful. For me, the form of a linear mixed model represented by Equation 2 in the R lme4 vignette better captures the joint combination of fixed and random effects in a mixed model.
