Let be a group under the operation and let be the algebra of formal Laurent series generated by the elements over the complex numbers. The elements of are of the form , where the coefficients are complex numbers and only a finite number of them with are nonzero. The left-shift on is the operator that corresponds to the multiplication with . The modified shift is defined by , if and .
With the operational calculus developed here the authors are finding solutions of , where is a polynomial, is a given element in , and is the unknown. If can be written as an element of , a solution in the abstract space can be written as , where is the solution of
Examples of this method are presented for the case of being a difference operator that acts on sequences, the case of being a differential operator , and the case of being the fractional differential operator .