A Generalization of Power and Alternating Power Sums to any Appell Polynomials

Dae San Kim, Taekyun Kim

The classical power sum and alternating power sum identities can be stated as \[ um^m_{i=0}s_n(i)=\frac1{n+1}(B_{n+1}(m+1)-B_{n+1}), \] \[ um^m_{i=0}(-1)^is_n(i)=\frac12((-1)^mE_n(m+1)+E_n), \] where $s_n(x)=x^n$ is the simplest possible Appell polynomial for the Sheffer pair$(1,t)$. The impetus for this research starts from the question that what if we replace $s_n(x)=x^n$ by any Appell polynomial. In this paper, we give a generalization of power and alternating power sums to any Appell polynomials.