Applied Umbral Calculus

Common ground to the three concepts are special polynomial sequences, called She er sequences. A polynomial sequence (sm (x))m2N0 is a sequence of polynomials sm (x) 2 K [x] such that deg sm = m, s0 6= 0. It is convenient to de ne sm = 0 for negative m. The coe cient ring K is assumed to be an integral domain. For this introduction to Finite Operator Calculus it su ces to choose K as R [!], the ring of real polynomials in some formal weight parameter !. A formal power series (t) 2 K [[x]] of order 1, i.e. , (0) = 0, 0 (0) invertible in K, will be called a delta series. We substitute the derivative operator D for t in a delta series, and obtain a shift-invariant linear operator Q on K [[x]] called a delta operator. The derivative D itself is a delta operator, and like D every delta operator