Delta operator

In mathematics, a delta operator is a shift-equivariant linear operator "formula_1" on the vector space of polynomials in a variable formula_2 over a field formula_3 that reduces degrees by one.
To say that formula_4 is shift-equivariant means that if formula_5, then
In other words, if "formula_7" is a "shift" of "formula_8", then "formula_9" is also a shift of "formula_10", and has the same "shifting vector" "formula_11".
To say that "an operator reduces degree by one" means that if "formula_7" is a polynomial of degree "formula_13", then "formula_9" is either a polynomial of degree formula_15, or, in case formula_16, "formula_9" is 0.
Sometimes a "delta operator" is defined to be a shift-equivariant linear transformation on polynomials in "formula_2" that maps "formula_2" 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 formula_20 has characteristic zero, since shift-equivariance is a fairly strong condition.
Basic polynomials.
Every delta operator "formula_4" 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.