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

Abstract

We prove a partial fraction decomposition of a quotient of two functions Eα(itx) and Iα(it) which are defined in terms of the Bessel functions Jα and Jα+1 of the first kind. This expansion leads naturally to the introduction of an orthonormal system with respect to the measure |x|2α+1dx2α+1Γ(α+1) in [−1,1], which we call the Fourier–Dunkl system of the second kind. Euler–Dunkl polynomials En,α(x) of degree n are defined by considering Eα(tx)∕Iα(t) as a generating function. It is shown that the sum ∑m=1∞1∕jm,α2k, where jm,α are the positive zeros of Jα, is equal (up to an explicit factor) to E2k−1,α(1). For α=1∕2 this leads to classical results of Euler since the function E1∕2(x) is the exponential function and En,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].