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Linear algebraic foundations of the operational calculi. (English) Zbl 1223.44003
Let $[p_k; k\in\Bbb Z[$ be a group under the operation $p_kp_n=p_{k+n}$ and let $\frak F$ be the algebra of formal Laurent series generated by the elements $p_k$ over the complex numbers. The elements of $\frak F$ 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 $\frak F$ 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 $k\ne 0$ 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 $\frak F$, and $f$ is the unknown. If $g$ can be written as an element of $\frak F$ , a solution in the abstract space $\frak F$ 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 $D^\alpha$.

44A45Classical operational calculus
44A40Calculus of Mikusiński and other operational calculi
13F25Formal power series rings
44A55Discrete operational calculus
26A33Fractional derivatives and integrals (real functions)
Full Text: DOI
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