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Eigenfunctions of Laguerre-type operators and generalized evolution problems. (English) Zbl 1097.34551
Setting $D=d/dx$, eigenfunctions of the following Laguerre-type operators $$D^{h+1}x^jD^{j-h}\text{ and } D^{p_1}x^{q_1}\cdots D^{p_r}x^{q_r}D^{p_{r+1}},$$ are explicitly given, where $h, j\ge 0, p_1, \cdots, p_{r+1}, q_1, \cdots, q_r>0$ are integers such that $j>h$, $p_1+\cdots+p_{r+1}=q_1+\cdots +q_r+1$. As an application, an operational solution of the evolution problem $$\hat{\Omega}_xS(x,t)=\frac{\partial}{\partial t} S(s, t)\quad \text{ in the half-plane } x>0,\quad K \lim_{x\to 0+}x^{-H}S(x,t)=s(t),$$ is obtained, where $\widehat\Omega_x$ is a differential operator in $x$, $H$, $K>0$ are constants and $s(t)$ is an analytic function.

34L40Particular ordinary differential operators
47E05Ordinary differential operators
Full Text: DOI
[1] Andrews, L. C.: Special functions of mathematics for engineers. (1998) · Zbl 0909.33002
[2] Dattoli, G.: Hermite-Bessel and Laguerre-Bessel functions: A by-product of the monomiality principle. Proc. of the workshop on ”advanced special functions and applications” (2000) · Zbl 1022.33006
[3] Appell, P.; De Fériet, J. Kampé: Fonctions hypergéométriques et hypersphériques. Polynômes d’hermite. (1926)
[4] Gould, H. W.; Hopper, A. T.: Operational formulas connected with two generalizations of Hermite polynomials. Duke math. J. 29, 51-62 (1962) · Zbl 0108.06504
[5] Srivastava, H. M.; Manocha, H. L.: A treatise on generating functions. (1984) · Zbl 0535.33001
[6] Ricci, P. E.: Tecniche operatoriali e funzioni speciali. (2001)
[7] G. Dattoli, P.E. Ricci and C. Cesarano, On a class of polynomials generalizing the Laguerre family, J. Comput. Anal. Appl. (to appear). · Zbl 1094.33007
[8] Dattoli, G.; Ricci, P. E.: Laguerre-type exponentials and the relevant L-circular and L-hyperbolic functions. Georgian math. J. 10, 481-494 (2003) · Zbl 1045.33001
[9] Riordan, J.: An introduction to combinatorial analysis. (1958) · Zbl 0078.00805
[10] Dattoli, G.; Ricci, P. E.; Pacciani, P.: Comments on the theory of Bessel functions with more than one index. Appl. math. Comput. 150, 603-610 (2004) · Zbl 1050.33004
[11] Bernardini, A.; Dattoli, G.; Ricci, P. E.: L-exponentials and higher order Laguerre polynomials. Proceedings of the fourth international conference of the society for special functions and their applications, 13-26 (2003) · Zbl 1077.33010
[12] C. Cesarano, B. Germano and P.E. Ricci, Laguerre-type Bessel functions, Integral Transforms Spec. Funct. (to appear). · Zbl 1080.33006
[13] Bretti, G.; Cesarano, C.; Ricci, P. E.: Laguerre-type exponentials and higher-order Appell polynomials. Computers math. Applic. 48, No. 5/6, 833-839 (2004) · Zbl 1072.33010
[14] Bretti, G.; Natalini, P.: Particular solutions for a class of ODE related to the L-exponential functions. Georgian math. J. 11, 59-67 (2004) · Zbl 1060.33002
[15] Dattoli, G.; Arena, A.; Ricci, P. E.: Laguerrian eigenvalue problems and wright functions. Mathl. comput. Modelling 40, No. 7/8, 877-881 (2004) · Zbl 1070.33009