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.
Other articles related to "operator, delta operator":
... are the group of Appell sequences, which are those sequences for which the operator Q is mere differentiation, and the group of sequences of binomial type, which are those that satisfy the identity A Sheffer ... are in the same such coset if and only if the operator Q described above – called the "delta operator" of that sequence – is the same linear operator in both cases ... (Generally, a delta operator is a shift-equivariant linear operator on polynomials that reduces degree by one ...
... 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 ...