Translativity of a summation method

The property of the method consisting in the preservation of summability of a series after adding to or deleting from it a finite number of terms. More precisely, a summation method $\mathcal{A}$ is said to be translative if the summability of the series $$ \sum_{k=0}^\infty a_k $$ to the sum $S_1$ implies that the series $$ \sum_{k=1}^\infty a_k $$ is summable by the same method to the sum $S_1 - a_0$, and conversely. For a summation method $\mathcal{A}$ defined by transformation of the sequence $S_n$ into a sequence or function, the property of translativity consists of the equivalence of the conditions $$ \mathcal{A}\text{-}\lim S_n = S $$ and $$ \mathcal{A}\text{-}\lim S_{n+1} = S $$

If the summation method is defined by a regular matrix $(A_{nk})$ (cf. Regular summation methods), then this means that $$\label{eq:a1} \lim_{n\rightarrow\infty} \sum_{k=0}^\infty A_{nk} S_k = S $$ always implies that $$\label{eq:a2} \lim_{n\rightarrow\infty} \sum_{k=0}^\infty A_{nk} S_{k+1} = S $$ and conversely. In cases when such an inference only holds in one direction, the method is called right translative if \ref{eq:a1} implies \ref{eq:a2} but the converse is false, or left translative if \ref{eq:a2} implies \ref{eq:a1} but the converse is false.

Many widely used summation methods have the property of translativity; for example, the Cesàro summation methods $(C,k)$ for $k > 0$, the Riesz summation method $R(n,k)$ for $k>0$ and the Abel summation method are translative; the Borel summation method is left translative.

References

[1]	R.G. Cooke, "Infinite matrices and sequence spaces" , Macmillan (1950)
[2]	S.A. Baron, "Introduction to the theory of summability of series" , Tartu (1966) (In Russian)