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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 -ν =1 Γ(ν) 0 exp(-at)t ν-1 dt,(1)

the author discusses a formal representation of a -ν f(x) with partial differential operators a as functions. In this way, by using the operator rule expλ xf(x)=f(x+λ), the formal representations of

α-y 2 x 2 -ν x n and1+y xx x -ν (-1) n x n n!

as polynomials ν H n (x,y) and ν 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 ν H n (x,y) and ν L n (x,y) are presented. Other applications of (1) are discussed. In particular, the formal representation of x x ν x 1-x 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 1/2 1/2 x must be understood as x 1/2 .

MSC:
33C45Orthogonal polynomials and functions of hypergeometric type
47A60Functional calculus of operators
33E20Functions defined by series and integrals
35R20Partial operator-differential equations