In mathematics, a difference operator maps a function, f(x), to another function, f(x + a) − f(x + b).

The forward difference operator

$\backslash Delta\; f(x)=f(x+1)-f(x)\backslash ,$

occurs frequently in the calculus of finite differences, where it plays a role formally similar to that of the derivative, but used in discrete circumstances. Difference equations can often be solved with techniques very similar to those for solving differential equations. Analogously we can have the backward difference operator

$\backslash nabla\; f(x)=f(x)-f(x-1).\backslash ,$

When restricted to polynomial functions f, the forward difference operator is a delta operator, i.e., a shift-equivariant linear operator on polynomials that reduces degree by 1.

n-th difference

---

The n-th forward difference of a function f(x) is given by

$\backslash Delta^n*(x)=\; \backslash sum\_\{k=0\}^n\; \{n\; \backslash choose\; k\}\; (-1)^\{n-k\}\; f(x+k)$

where $\{n\; \backslash choose\; k\}$ is the binomial coefficient. Forward differences applied to a sequence are sometimes called the binomial transform of the sequence, and, as such, have a number of interesting combinatorial properties.

Forward differences may be evaluated using the Nörlund-Rice integral. The integral representation for these types of series is interesting because the integral can often be evaluated using asymptotic expansion or saddle-point techniques; by contrast, the forward difference series can be extremely hard to evaluate numerically, because the binomial coefficients grow rapidly for large n.

Newton series

---

The Newton series or Newton forward difference equation is the relationship

$f(x+a)=\backslash sum\_\{k=0\}^\backslash infty\backslash frac\{\backslash Delta^k*(a)\}\{k!\}(x)\_k$

= \sum_{k=0}^\infty {x \choose k} \Delta^k ^*(a)

which holds for any polynomial function f and for some, but not all, analytic functions. Here,

$\{x\; \backslash choose\; k\}$

is the binomial coefficient, and

$(x)\_k=x(x-1)(x-2)\backslash cdots(x-k+1)$

is the "falling factorial" or "lower factorial" and the empty product (x)₀ defined to be 1. Note also the formal similarity of this result to Taylor's theorem; this is one of the observations that lead to the idea of umbral calculus.

In analysis with p-adic numbers, Mahler's theorem states that the assumption that f is a polynomial function can be weakened all the way to the assumption that f is merely continuous.

Carlson's theorem provides necessary and sufficient conditions for a Newton series to be unique, if it exists. However, a Newton series will not in general exist.

The Newton series, together with the Stirling series and the Selberg series, is a special case of the general difference series, all of which are defined in terms of scaled forward differences.

Rules for finding the difference applied to a combination of functions

---

Analogous to rules for finding the derivative, we have:

• Constant rule: If c is a constant, then

$\backslash triangle\; c\; =\; 0$

• Linearity: if a and b are constants,

$\backslash triangle\; (a\; f\; +\; b\; g)\; =\; a\; \backslash triangle\; f\; +\; b\; \backslash triangle\; g$

All of the above rules apply equally well to any difference operator, including $\backslash nabla$ as to $\backslash triangle$.

• Product rule:

$\backslash triangle\; (f\; g)\; =\; f\; \backslash triangle\; g\; +\; g\; \backslash triangle\; f\; +\; \backslash triangle\; f\; \backslash triangle\; g$

$\backslash nabla\; (f\; g)\; =\; f\; \backslash nabla\; g\; +\; g\; \backslash nabla\; f\; -\; \backslash nabla\; f\; \backslash nabla\; g$

• Quotient rule:

\det {\begin{vmatrix} g & \nabla g \\ 1 & 1 \end{vmatrix}}^{-1}

or

$\backslash nabla\backslash left(\; \backslash frac\{f\}\{g\}\; \backslash right)=\; \backslash frac\; \{g\; \backslash nabla\; f\; -\; f\; \backslash nabla\; g\}\{g\; \backslash cdot\; (g\; -\; \backslash nabla\; g)\}$

$\backslash triangle\backslash left(\; \backslash frac\{f\}\{g\}\; \backslash right)=\; \backslash frac\; \{g\; \backslash triangle\; f\; -\; f\; \backslash triangle\; g\}\{g\; \backslash cdot\; (g\; +\; \backslash triangle\; g)\}$

• Summation rules:

$\backslash sum\_\{n=a\}^\{b\}\; \backslash triangle\; f(n)\; =\; f(b+1)-f(a)$

$\backslash sum\_\{n=a\}^\{b\}\; \backslash nabla\; f(n)\; =\; f(b)-f(a-1)$

See also

---

• Newton polynomial
• Table of Newtonian series
• Lagrange polynomial

References

---

• Philippe Flajolet and Robert Sedgewick, "Mellin transforms and asymptotics: Finite differences and Rice's integrals", (1994) Theoretical Computer Science.

Finite differences | Mathematical analysis | Factorial and binomial topics

差分