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Linear algebraic foundations of the operational calculi. (English) Zbl 1223.44003

Let $\left[{p}_{k};k\in ℤ\left[$ be a group under the operation ${p}_{k}{p}_{n}={p}_{k+n}$ and let $𝔉$ be the algebra of formal Laurent series generated by the elements ${p}_{k}$ over the complex numbers. The elements of $𝔉$ are of the form $\sum {a}_{k}{p}_{k}$, where the coefficients ${a}_{k}$ are complex numbers and only a finite number of them with $k<0$ are nonzero. The left-shift on $𝔉$ is the operator ${S}_{-1}$ that corresponds to the multiplication with ${p}_{-1}$. The modified shift $L$ is defined by $L{p}_{k}={p}_{k-1}$, if $k\ne 0$ and $L{p}_{0}=0$.

With the operational calculus developed here the authors are finding solutions of $w\left(L\right)f=g$, where $w$ is a polynomial, $g$ is a given element in $𝔉$, and $f$ is the unknown. If $g$ can be written as an element of $𝔉$ , a solution in the abstract space $𝔉$ can be written as ${d}_{w}g$, where ${d}_{w}$ is the solution of $w\left(L\right){d}_{w}={p}_{0}$

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 $a\left(t\right)D+b\left(t\right)$, and the case of $L$ being the fractional differential operator ${D}^{\alpha }$.

##### MSC:
 44A45 Classical operational calculus 44A40 Calculus of Mikusiński and other operational calculi 13F25 Formal power series rings 44A55 Discrete operational calculus 26A33 Fractional derivatives and integrals (real functions)
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