In mathematics, a delta operator is a shift-equivariant linear operator
on the vector space of polynomials in a variable
that reduces degrees by one.
, and has the same "shifting vector"
To say that an operator reduces degree by one means that if
is a polynomial of degree
is either a polynomial of degree
Sometimes a delta operator is defined to be a shift-equivariant linear transformation on polynomials in
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.
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.