# Summation methods ordered by strength

A summation method is a partial function from scalar sequences to scalars, i.e. an element of the set $$\bigcup_{S \subseteq (\mathbb{N} \rightarrow \mathbb{C})} (S \rightarrow \mathbb{C})$$. A summation method $$\Sigma_1$$ is weaker than a summation method $$\Sigma_2$$ iff $$\Sigma_1 \subseteq \Sigma_2$$, i.e. $$\text{dom } \Sigma_1 \subseteq \text{dom } \Sigma_2$$ and $$\forall a \in \text{dom } \Sigma_1 : \Sigma_1(a) = \Sigma_2(a)$$. A summation method $$\Sigma_1$$ is consistent with a summation method $$\Sigma_2$$ iff $$\forall a \in \text{dom } \Sigma_1 \cap \text{dom } \Sigma_2 : \Sigma_1(a) = \Sigma_2(a)$$.

Cauchy summation is defined as the limit of partial sums, where such a limit exists: \begin{align} (\text{Cauchy}) \sum a &= \lim_{m \rightarrow \infty} \sum_{n=0}^m a_n \end{align}

A summation method is regular iff it is stronger than Cauchy summation. A summation method is linear iff $$\Sigma(sa) = s\Sigma(a)$$ and $$\Sigma(a+b) = \Sigma(a)+\Sigma(b)$$ for every scalar $$s$$, sequence $$a$$, and sequence $$b$$. A summation method is stable iff $$\Sigma(a)=a(0)+\Sigma(\sigma(a))$$, where $$\sigma(a)(n)=a(n+1)$$ is the shift operator. Is there a comprehensive list of summation methods (partially) ordered by strength? Are the following methods correctly ordered by strength?

\begin{align} (\text{Cesàro},\alpha) \sum a &= \lim_{m \rightarrow \infty} \sum_{n=0}^m \frac{\binom{m}{n}}{\binom{m+\alpha}{n}} a_n \\ (\text{Lambert}) \sum a &= \lim_{\varepsilon \rightarrow 0^+} \lim_{m \rightarrow \infty} \sum_{n=0}^m a_n \frac{\varepsilon(n+1) \mathrm{e}^{-\varepsilon(n+1)}}{1-\mathrm{e}^{-\varepsilon(n+1)}} \\ (\text{Abelian means},\lambda) \sum a &= \lim_{\varepsilon \rightarrow 0^+} \lim_{m \rightarrow \infty} \sum_{n=0}^m a_n \exp(-\varepsilon \lambda_n) \\ (\text{Borel},\alpha) \sum a &= \int_0^\infty \mathrm{e}^{-t} \lim_{m \rightarrow \infty} \sum_{n=0}^m \frac{a_n t^{n\alpha}}{(n\alpha)!} \,\mathrm{d}t \end{align}

Note that Abel and Lindelöf summation are Abelian means summation for $$\lambda_n = n$$ and $$\lambda_n = n \log n$$, respectively. Do the following methods fall somewhere in the linear order above?

\begin{align} (\text{Le Roy}) \sum a &= \lim_{z \rightarrow 1^-} \lim_{m \rightarrow \infty} \sum_{n=0}^m \frac{(zn)!}{n!} a_n \\ (\text{Mittag-Leffler}) \sum a &= \lim_{\varepsilon \rightarrow 0} \lim_{m \rightarrow \infty} \sum_{n=0}^m \frac{a_n}{(\varepsilon n)!} \end{align}