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.

Read more about Delta Operator: Examples, Basic Polynomials

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