I would like to Taylor expand a function of multiple operators (which may not commute). For a single operator $A$, I have seen a function $f(A)$ expanded as
$$ f(A) = f(0) + A f'(0) + \frac{A^2}{2} f''(0) + \cdots. $$
For a function of two variables $f(x,y)$, the "usual" Taylor expansion would be
$$ f(x,y) = f(0,0) + x \frac{\partial f}{\partial x} (0,0) + y \frac{\partial f}{\partial y}(0,0) + \frac{x^2}{2} \frac{\partial^2 f}{\partial x^2} (0,0) + xy \frac{\partial^2 f}{\partial x \partial y} (0,0) + \frac{y^2}{2} \frac{\partial^2 f}{\partial y^2} (0,0) + \cdots. $$
If the inputs to this function are instead operators $A$ and $B$, how can one Taylor expand $f(A,B)$? The cross term $xy$ in the above expansion is problematic because $A$ and $B$ may not commute.
I am new to functional calculus. The discussion I have found on sites such as Wikipedia (Functional calculus and other referenced pages) seems confined to functions of a single operator. If someone could point me to a good reference on this question, I would really appreciate it!