Playing around with the Airy equation $$y''=t y$$ I came across determinant expressions of the form $$\text{Ai}(x) \text{Bi}(y)-\text{Ai}(y) \text{Bi}(x), $$ as well as $$ \text{Ai}(x) \text{Bi}'(y)-\text{Ai}'(y) \text{Bi}(x).$$
I was wondering if there exist any "Airy identities", similarly to the trigonometric case, which could help me simplify such expressions.
Thank you!