Schwarz differential

The principal part of the Schwarz symmetric derivative of order $n$. More precisely, if for a function $ f $ of a real variable,

$$ \Delta ^ {n} f ( x, \Delta x) = \sum_{k=0}^ { n } \left ( \begin{array}{c} n \\ k \end{array} \right ) (- 1) ^ {k} f \left ( x + n- \frac{2k}{2} \Delta x \right ) = $$

$$ = \ A \cdot ( \Delta x) ^ {n} + o(( \Delta x) ^ {n} ), $$

then the expression $ A \cdot ( \Delta x) ^ {n} $ is called the Schwarz differential of order $ n $. When a Schwarz differential is mentioned without specifying the order, it is usually assumed that $ n= 2 $.