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 with partial differential operators as functions. In this way, by using the operator rule , the formal representations of
as polynomials and of and , 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 and are presented. Other applications of (1) are discussed. In particular, the formal representation of 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 must be understood as .