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, since shift-equivariance is a fairly strong condition.