Symmetric difference of order n

at a point $ x $ of a function $ f $ of a real variable

The expression

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

The following expression is often also referred to as a symmetric difference:

$$ \sum _ {k = 0 } ^ { n } \left ( \begin{array}{c} n \\ k \end{array} \right ) (- 1) ^ {k} f ( x + ( n - 2k) h). $$

It is obtained from the above by substituting $ 2h $ for $ h $. If $ f ( x) $ has an $ n $- th order derivative $ f ^ { ( n) } ( x) $ at $ x $, then

$$ \Delta _ {s} ^ {n} f ( x, h) = \ f ^ { ( n) } ( x) h ^ {n} + o ( h ^ {n} ). $$

References

[a1]	H. Meschkowski, "Differenzengleichungen" , Vandenhoeck & Ruprecht (1959)
[a2]	L.N. Milne-Thomson, "The calculus of finite differences" , Chelsea, reprint (1981)
[a3]	N.E. Nörlund, "Volesungen über Differenzenrechnung" , Chelsea, reprint (1954)