Appell-Dunkl sequences and Hurwitz-Dunkl zeta functions

Abstract

Dunkl theory on the real line involves some tools such as the Dunkl derivativeΛαf(x)=ddxf(x)+2α+12f(x)−f(−x)x or the Dunkl exponential Eα(z) that is defined in terms of the Bessel functions. Taking α=−1/2 we get Λ−1/2=d/dx and E−1/2(z)=ez, hence, the classic derivative and exponential are particular cases. In recent years, some papers have generalized, in a Dunkl sense, number theoretic concepts such as Appell sequences, and then they are called Appell-Dunkl sequences; in particular, the so called Bernoulli-Dunkl and Euler-Dunkl polynomials have been defined, among others. Here we generalize, also in a Dunkl sense, some Hurwitz or Lerch zeta functions such as ζ(s,x)=∑n=0∞1/(n+x)s and, in addition, we get properties that relate those functions, extended to the s-complex plane and evaluated at negative integers s, with Bernoulli-Dunkl and Euler-Dunkl polynomials. One of the results we get for the “Dunkl zeta function” ζα(s) isζα(1−s)=Γ(s)cos⁡(πs2)∑n=1∞1sns,Re(s)>1 (where sn are the positive zeros of the Bessel function Jα+1(x)). This equation provides a generalization of the reflection formula of the Riemann zeta function, where the function ∑n=1∞1/sns is playing a similar role as ∑n=1∞1/ns.