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DifferenceDelta

Status: Stable

documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

Description

DifferenceDelta[f, i] gives the forward difference (f /. i -> i+1) - f, the discrete analogue of D. It is the left inverse of indefinite Sum.

Examples

All examples below are verified against the current Mathilda build.

In[1]:= DifferenceDelta[i^2, i]
Out[1]= 1 + 2 i

In[2]:= DifferenceDelta[Sum[k k!, k], k]
Out[2]= -Factorial[k] + Factorial[1 + k]

Implementation notes

Algorithm. DifferenceDelta[f, i] computes the forward difference (f /. i -> i+1) - f, the discrete analogue of D and the left inverse of indefinite Sum. builtin_differencedelta (src/sum/sum_gosper.c) requires the second argument to be a symbol, substitutes i -> i+1 into f (shift_var, implemented via ReplaceAll), subtracts the original f, and returns the Expand-ed result. Returns NULL (unevaluated) unless the variable is a symbol.

Data structures. Plain Expr* tree manipulation built on the existing ReplaceAll, subtraction, and Expand builtins (shift_var, sum_sub, sum_eval). No closed-form machinery — it is a thin structural operator that lives alongside Gosper's summation because the two are inverse operations.

Attributes: Protected.

Implementation status

Stable — documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

References

Notes & additional examples

Worked examples

In[1]:= DifferenceDelta[n^2, n]
Out[1]= 1 + 2 n

In[1]:= DifferenceDelta[f[n], n]
Out[1]= -f[n] + f[1 + n]

In[1]:= DifferenceDelta[n^3, n]
Out[1]= 1 + 3 n + 3 n^2

In[1]:= DifferenceDelta[Binomial[n, k], n]
Out[1]= -Binomial[n, k] + Binomial[1 + n, k]

Notes

DifferenceDelta[f, n] is the forward difference operator Δ f = (f /. n -> n+1) - f, the discrete analogue of the derivative D. On n^2 it returns 2 n + 1, the discrete counterpart of 2 n; applied to n^3 it gives 3 n^2 + 3 n + 1. For an unknown function head it expands literally to f[n+1] - f[n]. The fourth example is Pascal's rule in disguise: Binomial[n+1, k] - Binomial[n, k] = Binomial[n, k-1]. DifferenceDelta is the left inverse of indefinite Sum, mirroring the way D inverts the indefinite integral.