deltaofdeltas

Deltaofdeltas is a term sometimes used in mathematics and numerical analysis to refer to the second application of the forward-difference operator, i.e., the second finite difference. If Δ is the forward-difference operator defined by Δ f(n) = f(n+1) − f(n), then Δ^2 f(n) = Δ(Δ f(n)) = f(n+2) − 2 f(n+1) + f(n). The expression can also be written as Δ^2 f(n) or ΔΔ f(n).

A simple example: for f(n) = n^2, Δ f(n) = (n+1)^2 − n^2 = 2n + 1, and Δ^2 f(n) = Δ(2n + 1) = 2. Thus the second finite difference is constant for a quadratic sequence.

Deltaofdeltas, via Δ^2 f(n), measures discrete concavity or curvature of a sequence and is constant for polynomial sequences of degree two. It has practical use in numerical analysis and related fields, including numerical differentiation, interpolation (such as Newton’s forward interpolation that uses forward differences), and the solution of difference equations. In time series analysis, the second difference is used to assess trend changes and stationarity.

Relation to the continuous case: for a function f with step size h, Δ^2_h f(x) approximates h^2 f''(x); with h = 1, Δ^2 f(n) provides a discrete approximation to the second derivative. Generalizations exist as higher-order differences Δ^k, with Δ^2 being the first nontrivial case beyond the first difference.

Notes: deltaofdeltas is not a widely standardized term. In formal writing, it is common to refer to the second finite difference as Δ^2 f or ΔΔ f rather than using the coined phrase. See also finite difference, forward difference, and Newton’s forward interpolation.