A matrix approach to Sheffer polynomials
The purpose of the talk is to present an extension of the matrix representation of Appell polynomials, done by L. Aceto et al. (L. Aceto, H.R. Malonek, G. Tomaz, A unified matrix approach to the representation of Appell polynomials, Integral Transforms Spec. Funct. 26 (2015) 426–441) to the more general class of the Sheffer polynomials. The core of the proposed approach is the so-called creation matrix, a special subdiagonal matrix having as nonzero entries positive integer numbers. We show that the Sheffer polynomials can be represented by two matrices, both connect to it, and that different choices of these matrices determine different classes of Sheffer polynomials.
Fundamental solution of the time-fractional telegraph equation in higher dimensions
In this talk we present some results concerning the fundamental solution (FS) of the multidimensional time-fractional telegraph equation with time-fractional derivatives in the Caputo sense. In the Fourier domain the FS is expressed in terms of a multivariate Mittag-Leffler function. The Fourier inversion leads to a representation of the FS in terms of a H-function of two variables. An explicit series representation of the FS, depending on the parity of the dimension, is presented. As an application, we study a telegraph process with Brownian time. Finally, we present some moments of integer order of the FS, and some plots of the FS for some particular values of the dimension and of the fractional parameters.