Delta operator

In mathematics, a delta operator is a shift-equivariant linear operator Q on the vector space of polynomials in a variable x that reduces degrees by one.

To say that Q is shift-equivariant means that if f(x) = g(x + a), i.e., f is a "shift" of g, then (Qf)(x) = (Qg)(x + a), Qf is the same shift of Qg that f is of g. That the operator reduces degrees by one means that if f is a polynomial of degree n, then Qf is either a polynomial of degree n - 1, or, in case n = 0, Qf is 0.

Sometimes a delta operator is defined to be a shift-equivariant linear transformation on polynomials in x that maps x to a nonzero constant. Seemingly weaker than the definition given above, this latter characterization can be shown to be equivalent to the stated definition, since shift-equivariance is a fairly strong condition.

The forward difference operator (Δf)(x) = f(x + 1) - f(x) is a delta operator. Differentiation with respect to x, written as D, is also a delta operator. Any operator of the form

<math>\sum_{k=1}^\infty c_k D^k</math>

where c₁ is not 0, can readily be seen to be a delta operator. It can be shown that there are no other delta operators than those that can be written in this form. For example, the difference operator given above can be expanded as

<math>\Delta=e^D-1=\sum_{k=1}^\infty \frac{D^k}{k!}.</math>

Every delta operator Q has a unique sequence of "basic polynomials", a polynomial sequence defined by three conditions:

<math>p_0(x)=1,</math>

<math>p_{n}(0)=0,</math>

<math>(Qp_n)(x)=np_{n-1}(x),</math>

for every positive integer n. Such a sequence of basic polynomials is always of binomial type, and it can be shown that no other sequences of binomial type exist. If the first two conditions above are dropped, then the third condition says this polynomial sequence is a Sheffer sequence -- a more general concept.

The name "delta operator" is due to F. Hildebrandt[?].