*(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

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 $exp\left(\lambda \frac{\partial}{\partial x}\right)f\left(x\right)=f(x+\lambda )$, the formal representations of

as polynomials ${}_{\nu}{H}_{n}(x,y)$ and ${}_{\nu}{L}_{n}(x,y)$ 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 ${}_{\nu}{H}_{n}(x,y)$ and ${}_{\nu}{L}_{n}(x,y)$ 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 |