Matrix summation method
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A matrix summation method is one of the methods for summing series and sequences using an infinite matrix. Employing an infinite matrix $[a_{nk}]$, $n,k=1,2,\ldots,$ a given sequence $(s_n)$ is transformed into the sequence $\sigma_n$:
$$
\sigma_n = \sum_{k=1}^\infty a_{nk}s_k.
$$
If the series on the right-hand side converges for all $n=1,2,\ldots,$ and if the sequence $\sigma_n$ has a limit $s$ for $n \rightarrow \infty$:
$$
\lim_{n\rightarrow\infty} \sigma_n = s,
$$
then the sequence $(s_n)$ is said to be summable by the method determined by the matrix $[a_{nk}]$, or simply summable by the matrix $[a_{nk}]$, and the number $s$ is referred to as its limit in the sense of this summation method. If $(s_n)$ is regarded as the sequence of partial sums of a series
\begin{equation}
\label{eq1}
\sum_{k=1}^\infty u_k,
\end{equation}
then this series is said to be summable to the sum $s$ by the matrix
$[a_{nk}]$.
A matrix summation method for series can be also defined directly by transforming the series \ref{eq1} into a sequence $(\gamma_n)$: \begin{equation} \label{eq2} \gamma_n = \sum_{k=1}^\infty g_{nk}u_k, \end{equation} where $[g_{nk}]$ is a given matrix. In this case the series \ref{eq1} is said to be summable to the sum $s$ if, for all $n=1,2,\ldots,$ the series on the right-hand side in \ref{eq2} converges and $$ \lim_{n\rightarrow\infty} \gamma_n = s, $$
Less often used are matrix summation methods defined by a transformation of a series \ref{eq1} into a series \begin{equation} \label{eq3} \sum_{n=1}^\infty \alpha_n, \end{equation} where $$ \alpha_n = \sum_{k=1}^\infty \alpha_{nk}u_k, $$ or by a transformation of a sequence $(s_n)$ into a series \begin{equation} \label{eq4} \sum_{n=1}^\infty \beta_n, \end{equation} where $$ \beta_n = \sum_{k=1}^\infty \beta_{nk}s_k, \quad n=1,2,\ldots, $$ which use matrices $[\alpha_{nk}]$ and $[\beta_{nk}]$, respectively. In these cases the series \ref{eq1} with the partial sums $s_n$ is summable to the sum $s$ if the series \ref{eq3} converges to $s$ or, respectively, if the series \ref{eq4} converges to $s$.
The matrix of a summation method all entries of which are non-negative is called a positive matrix. Among the matrix summation methods one finds, for example, the Voronoi summation method, the Cesàro summation methods, the Euler summation method, the Riesz summation method $(R,p_n)$, the Hausdorff summation method, and others (see also Summation methods).
References
| [1] | S.A. Baron, "Introduction to the theory of summability of series", Tartu (1966) (In Russian) |
| [2] | R.G. Cooke, "Infinite matrices and sequence spaces", Macmillan (1950) |
| [3] | G.H. Hardy, "Divergent series", Clarendon Press (1949) |
| [4] | G.P. Kangro, "Theory of summability of sequences and series" J. Soviet Math., 5 : 1 (1976) pp. 1–45 Itogi Nauk. i Tekhn. Mat. Anal., 12 (1974) pp. 5–70 |
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