Fourier-Dunkl system of the second kind and Euler-Dunkl polynomials (Q2315025)

In mathematics, the Euler polynomials are a classical mathematical tool, which occur frequently in number theory and have various application in probability, special polynomials, \(q\)-calculus and so on. In this paper, the authors proved a partial fraction decomposition of a quotient of two functions \(E_{\alpha}(itx)\) and \(I_{\alpha}(it)\) which are defined in terms of the Bessel functions \(J_{\alpha}\) and \(J_{\alpha+1}\) of the first kind. The Fourier-Dunkl system of the second kind and Euler-Dunkl polynomials have been studied. Furthermore, a sampling theorem of Whittaker-Shannon-Kotel'nikov type is established which is strongly related to the above-mentioned partial decomposition and which holds for all functions in the Payley-Wiener space defined by the Dunkl transform in \([-1,1]\).