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Zbl 1099.33006
Dattoli, G.
Operational methods, fractional operators and special polynomials.
(English)
[J] Appl. Math. Comput. 141, No. 1, 151-159 (2003). ISSN 0096-3003

The paper is devoted to application of fractional powers of operators to special functions and operator-differential equations. Using the operator formula $$a^{-\nu}={\frac{1}{\Gamma (\nu)}}\int^{\infty}_{0}\exp (-at)t^{\nu -1}dt,\eqno(1)$$ the author discusses a formal representation of $a^{-\nu}f(x)$ with partial differential operators $a$ as functions. In this way, by using the operator rule $\exp \left(\lambda{\frac{\partial}{\partial x}}\right)f(x)=f(x+\lambda)$, the formal representations of $$\left(\alpha -y{\frac{\partial^{2}}{\partial x^{2}}}\right)^{-\nu}x^{n} \text{ and } \left(1+y{\frac{\partial}{\partial x}}x{\frac{\partial}{\partial x}}\right)^{-\nu} \left[{\frac{(-1)^{n}x^{n}}{n!}}\right]$$ 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 {\it 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. \par Note. In the formula (46) of the paper the relation ${\frac{\partial^{1/2}}{\partial^{1/2}x}}$ must be understood as $\left({\frac{\partial}{\partial x}}\right)^{1/2}$.
[Anatoliy Aleksandrovich Kilbas (Minsk)]
MSC 2000:
*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

Keywords: fractional powers of differential operators; modified Hermite and Laguerre polynomials; Riemann zeta function; partial operator-differential equation

Citations: Zbl 1022.33006

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