Asymptotics of Series Arising from the Approximation of Periodic Functions by Riesz and Cesàro Means

Asymptotic expansions in powers of $\delta$ as $\delta\to+\infty$ of the series
$$ \sum_{k=0}^\infty(-1)^{(\beta+1)k}\frac{Q((\delta^\alpha-(ak+b)^\alpha)_+)}{(ak+b)^{r+1}}, $$
where $\beta\in\mathbb Z$, $\alpha,a,b>0$, and $r\in\mathbb C$, while $Q$ is an algebraic polynomial satisfying the condition $Q(0)=0$, are obtained. In special cases, these series arise from the approximation of periodic differentiable functions by the Riesz and Cesàro means.