A Composition Theorem for Binomially Weighted Averages

We study binomially weighted summation methods given by $$(x_n{)}_{n\in \mathbb{N}}\mapsto {(\sum _{k=0}^n\left(\genfrac{}{}{0px}{}{n}{k}\right)r^k(1-r{)}^{n-k}{x}_k)}_{n\in \mathbb{N}}$$ for $r\in (0,1)$, and their behavior under composition with summation methods of the form $$(x_n{)}_{n\in \mathbb{N}}\mapsto {\left(\sum _{k=0}^n{\lambda }_k{x}_{n-k}\right)}_{n\in \mathbb{N}}\mathrm{.}$$ Our main result shows that if the binomially weighted averages of a sequence $(x_n{)}_{n\in \mathbb{N}}$ converge to a limit then the binomially weighted averages of the sequence ${\left({\sum }_{k=0}^n{\lambda }_k{x}_{n-k}\right)}_{n\in \mathbb{N}}$ converge to the same limit whenever $({\lambda }_n{)}_{n\in \mathbb{N}}$ is an absolutely summable sequence with ${\sum }_{k=0}^{\mathrm{\infty }}{\lambda }_k=1$. This result disproves a theorem appearing in the literature. Additionally, we discuss applications and extensions of our main result to compositions with weighted Cesàro averages.