series. Another way of generalization is making coefficients depend on point : , thus considering weighted finite difference. Also one may make the step depend on point : . Such generalizations are useful for constructing different modulus of continuity. The generalized difference can be seen as the polynomial rings . It leads to difference algebras. Difference operator generalizes to Möbius inversion over a partially ordered set. As a convolution operator: Via the formalism of incidence algebras, difference operators and other Möbius inversion can be represented by convolution with a function on the poset, called the Möbius function ; for the difference operator, is the sequence . Multivariate finite differences Finite differences can be considered in more than one variable. They are analogous to partial derivatives in several variables. Some partial derivative approximations are: Alternatively, for applications in which the computation of is the most costly step, and both first and second derivatives must be computed, a more efficient formula for the last case is since the only values to compute that are not already needed for the previous four equations are and . See also References Richardson, C. H. (1954): An Introduction to the Calculus of Finite Differences (Van Nostrand (1954) online copy Mickens, R. E. (1991): Difference Equations: Theory and Applications'' (Chapman and Hall/CRC) External links Table of useful finite difference formula generated using Mathematica D. Gleich (2005), Finite Calculus: A Tutorial for Solving Nasty Sums Discrete Second Derivative from Unevenly Spaced Points Numerical differential equations Mathematical analysis Factorial and binomial topics Linear operators in calculus Numerical analysis Non-Newtonian calculus