zbMATH — the first resource for mathematics

Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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)