Difference operator

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

The forward difference operator

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

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

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

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

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

where ${n \choose k}$ 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, named after Isaac Newton, is the relationship

$f(x+a)=\sum _{k=0}^{\infty }{\frac {\Delta ^{k}[f](a)}{k!}}(x)_{k}=\sum _{k=0}^{\infty }{x \choose k}\Delta ^{k}[f](a)$

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

${x \choose k}$

is the binomial coefficient, and

$(x)_{k}=x(x-1)(x-2)\cdots (x-k+1)$

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

$\triangle c=0$

• Linearity: if a and b are constants,

$\triangle (af+bg)=a\triangle f+b\triangle g$

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

• Product rule:

$\triangle (fg)=f\triangle g+g\triangle f+\triangle f\triangle g$

$\nabla (fg)=f\nabla g+g\nabla f-\nabla f\nabla g$

• Quotient rule:

$\nabla \left({\frac {f}{g}}\right)={\frac {1}{g}}\det {\begin{bmatrix}\nabla f&\nabla g\\f&g\end{bmatrix}}\det {\begin{bmatrix}g&\nabla g\\1&1\end{bmatrix}}^{-1}$

or

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

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

• Summation rules:

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

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

See also

• Newton polynomial
• Table of Newtonian series
• Lagrange polynomial
• Gilbreath's conjecture

References

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