I am searching for a general formula for directly calculating the second, fourth, and sixth derivative directly from a time series data. Wikipedia has a formula for finding an $n$th order central finite difference. I have searched a lot of places but I cannot find it in any reference book? The equation is given on a Wikipedia page under the section "Higher Order Difference."
https://en.wikipedia.org/wiki/Finite_difference
Does anyone know a good reference where one can find a general formula for nth order finite difference?
Wikipedia articles come with extensive lists of references. Just picking one promising-looking reference at random (Calculus of Finite Differences by Jordan) I found a section on central differences including the general formulas for $n$th order central differences (they're written in terms of other formulas for forward differences, but putting the pieces together is straightfoward).