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}


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