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Consider the ring $\mathbb{C}[[X,Y]]$ and its subring $\mathbb{C}[[X+iY]]$, where $i=\sqrt{-1}$. One can show that $f(X,Y):=u(X,Y)+iv(X,Y)\in \mathbb{C}[[X,Y]]$ lies in $\mathbb{C}[[X+iY]]$ iff $u$ ...

Let. $$f := (x, y) \mapsto \text{BesselK}(1, c \cdot (a - b \cdot (x + y))) \cdot \exp(c \cdot b \cdot (y - x))$$ Is there a close formula for this $$\frac{\partial^{m+n}}{\partial y^m \partial x^n} f(...