Delta operator

In mathematics, a delta operator is a shift-equivariant linear operator on the vector space of polynomials in a variable over a field that reduces degrees by one.
To say that is shift-equivariant means that if, then
In other words, if is a "shift" of, then is also a shift of, and has the same "shifting vector".
To say that an operator reduces degree by one means that if is a polynomial of degree, then is either a polynomial of degree, or, in case, is 0.
Sometimes a delta operator is defined to be a shift-equivariant linear transformation on polynomials in that maps to a nonzero constant. Seemingly weaker than the definition given above, this latter characterization can be shown to be equivalent to the stated definition when has characteristic zero, since shift-equivariance is a fairly strong condition.

Examples

The forward difference operator
Differentiation with respect to x, written as D, is also a delta operator.
Any operator of the form
The generalized derivative of time scale calculus which unifies the forward difference operator with the derivative of standard calculus is a delta operator.
In computer science and cybernetics, the term "discrete-time delta operator" is generally taken to mean a difference operator
Basic polynomials

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

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.

Delta operator

Examples

Basic polynomials