Fourier-Dunkl system of the second kind and Euler-Dunkl polynomials

We prove 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. This expansion leads naturally to the introduction of an orthonormal system with respect to the measure in vertical bar x vertical bar(2 alpha+1)dx/2(alpha+1)Gamma(alpha+1) in [-1, 1], which we call the Fourier-Dunkl system of the second kind. Euler-Dunkl polynomials epsilon(n,alpha)(x) of degree n are defined by considering E-alpha(tx)/I-alpha(t) as a generating function. It is shown that the sum Sigma(infinity)(m=1) 1/j(m,alpha)(2k), where jm,alpha are the positive zeros of J alpha, is equal (up to an explicit factor) to epsilon(2k-1), (alpha)(1). For alpha = 1/2 this leads to classical results of Euler since the function E-1/2(x) is the exponential function and epsilon(n,1/2)(x) are (essentially) the Euler polynomials. In the second part of the paper 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]. (C) 2019 Elsevier Inc. All rights reserved.