# 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)
Operational methods, fractional operators and special polynomials. (English) Zbl 1099.33006

The paper is devoted to application of fractional powers of operators to special functions and operator-differential equations. Using the operator formula

${a}^{-\nu }=\frac{1}{{\Gamma }\left(\nu \right)}{\int }_{0}^{\infty }exp\left(-at\right){t}^{\nu -1}dt,\phantom{\rule{2.em}{0ex}}\left(1\right)$

the author discusses a formal representation of ${a}^{-\nu }f\left(x\right)$ with partial differential operators $a$ as functions. In this way, by using the operator rule $exp\left(\lambda \frac{\partial }{\partial x}\right)f\left(x\right)=f\left(x+\lambda \right)$, the formal representations of

${\left(\alpha -y\frac{{\partial }^{2}}{\partial {x}^{2}}\right)}^{-\nu }{x}^{n}\phantom{\rule{4.pt}{0ex}}\text{and}\phantom{\rule{4.pt}{0ex}}{\left(1+y\frac{\partial }{\partial x}x\frac{\partial }{\partial x}\right)}^{-\nu }\left[\frac{{\left(-1\right)}^{n}{x}^{n}}{n!}\right]$

as polynomials ${}_{\nu }{H}_{n}\left(x,y\right)$ and ${}_{\nu }{L}_{n}\left(x,y\right)$ of $x$ and $y$, are deduced. These constructions are modifications of polynomials connected with the classical Hermite and Laguerre polynomials; see G. Datolli [Advanced special functions and applications. Proceedings of the workshop, Melfi, Italy, May 9–12, 1999. Rome: Aracne Editrice. Proc. Melfi Sch. Adv. Top. Math. Phys. 1, 147–164 (2000; Zbl 1022.33006)]. Some properties of ${}_{\nu }{H}_{n}\left(x,y\right)$ and ${}_{\nu }{L}_{n}\left(x,y\right)$ are presented. Other applications of (1) are discussed. In particular, the formal representation of ${\left(x\frac{\partial }{\partial x}\right)}^{\nu }\left[\frac{x}{1-x}\right]$ as the Riemann zeta function is given, and a formal solution of the Cauchy problem for one partial operator-differential equation is obtained.

Note. In the formula (46) of the paper the relation $\frac{{\partial }^{1/2}}{{\partial }^{1/2}x}$ must be understood as ${\left(\frac{\partial }{\partial x}\right)}^{1/2}$.

##### MSC:
 33C45 Orthogonal polynomials and functions of hypergeometric type 47A60 Functional calculus of operators 33E20 Functions defined by series and integrals 35R20 Partial operator-differential equations