To say that Q is shiftequivariant means that if f(x) = g(x + a), i.e., f is a "shift" of g, then (Qf)(x) = (Qg)(x + a), Qf is the same shift of Qg that f is of g. That the operator reduces degrees by one means that if f is a polynomial of degree n, then Qf is either a polynomial of degree n  1, or, in case n = 0, Qf is 0.
Sometimes a delta operator is defined to be a shiftequivariant linear transformation on polynomials in x that maps x 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 shiftequivariance is a fairly strong condition.
The forward difference operator (Δf)(x) = f(x + 1)  f(x) is a delta operator. Differentiation with respect to x, written as D, is also a delta operator. Any operator of the form
Every delta operator Q has a unique sequence of "basic polynomials", a polynomial sequence defined by three conditions:
The name "delta operator" is due to F. Hildebrandt[?].
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