In mathematics, a delta operator is a shift-equivariant linear operator on the vector space of polynomials in a variable x over a field \mathbb{K} that reduces degrees by one. To say that Q is shift-equivariant means that if, then In other words, if f is a "shift" of g, then Qf is also a shift of Qg, and has the same "shifting vector" a. To say that an operator reduces degree 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 when \mathbb{K} has characteristic zero, since shift-equivariance is a fairly strong condition.

Examples

Basic polynomials

Every delta operator Q 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.