Definition
Fix a polynomial sequence pn. Define a linear operator Q on polynomials in x by
This determines Q on all polynomials. The polynomial sequence pn is a Sheffer sequence if the linear operator Q just defined is shift-equivariant. Here, we define a linear operator Q on polynomials to be shift-equivariant if, whenever f(x) = g(x + a) = Ta g(x) is a "shift" of g(x), then (Qf)(x) = (Qg)(x + a); i.e., Q commutes with every shift operator: TaQ = QTa. Such a Q is a delta operator.
Read more about this topic: Sheffer Sequence
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