Linear algebraic foundations of the operational calculi
Let \([p_k; k\in\mathbb Z[\) be a group under the operation \(p_kp_n=p_{k+n}\) and let \(\mathfrak F\) be the algebra of formal Laurent series generated by the elements \(p_k\) over the complex numbers. The elements of \(\mathfrak F\) are of the form \(\sum a_kp_k\), where the coefficients \(a_k\) are complex numbers and only a finite number of them with \(k<0\) are nonzero. The left-shift on \(\mathfrak F\) is the operator \(S_{-1}\) that corresponds to the multiplication with \(p_{-1}\). The modified shift \(L\) is defined by \(Lp_k=p_{k-1}\), if \(k\neq 0\) and \(Lp_0=0\). With the operational calculus developed here the authors are finding solutions of \(w(L)f=g\), where \(w\) is a polynomial, \(g\) is a given element in \(\mathfrak F\), and \(f\) is the unknown. If \(g\) can be written as an element of \(\mathfrak F\) , a solution in the abstract space \(\mathfrak F\) can be written as \(d_wg\), where \(d_w\) is the solution of \(w(L)d_w=p_0\) Examples of this method are presented for the case of \(L\) being a difference operator \(\Delta\) that acts on sequences, the case of \(L\) being a differential operator \(a(t)D+b(t)\), and the case of \(L\) being the fractional differential operator \(D^\alpha\).
scientific article; zbMATH DE number 5919051