Sheffer-Dunkl Sequences Via Umbral Calculus in the Dunkl Context

Abstract Umbral calculus refers to a series of techniques that can be used to prove some polynomial formulas. Nowadays, it mostly involves the study of Sheffer sequences. In this paper, we focus on a generalization of umbral calculus in a Dunkl context (that we call Dunkl-umbral calculus). Here, the operators of classical umbral calculus are naturally replaced by operators of the Dunkl theory on the real line. In this context we define for the first time Sheffer-Dunkl sequences, $$\{s_{n,\alpha }(x)\}_{n=0}^\infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mrow> <mml:mo>{</mml:mo> <mml:msub> <mml:mi>s</mml:mi> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>,</mml:mo> <mml:mi>α</mml:mi> </mml:mrow> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>}</mml:mo> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:mi>∞</mml:mi> </mml:msubsup> </mml:math> , and provide some properties and examples. We also connect, via Dunkl-umbral calculus, properties of some polynomials in a Dunkl sense that have appeared in the literature in the recent years, like Bernoulli-Dunkl or Euler-Dunkl polynomials.