TITLE: What can be computed by axiomatic summation?
QUESTION [7 upvotes]: Here are three simple properties one might require of a summation method for divergent series: 

A stable summation scheme is one in which (assuming also each sums are defined iff the other is) $$\sum_{n=1}^\infty a_n = a_1 + \sum_{n=1}^\infty a_{n+1}$$
A linear one satisfies (whenever the RHS is defined) $$\sum_{n=1}^\infty (\kappa a_n+b_n) = \kappa\sum_{n=1}^\infty a_n + \sum_{n=1}^\infty b_n$$(Edit: Added $\kappa$, that was forgetfulness.)
A regular one agrees with normal summation when both are defined. (That is, if one has a convergent series, it is given its normal value by the summation method.)

If one assumes these rules, what class of series may be computed? It sounds like it should be the set of series which differ by a convergent series from another which is periodic up to rescaling; i.e. $a_n = c_n + k^n p_n$ where $\sum c_n$ is convergent and $p_{n+m} = p_n$ for some $m$. Is that correct? (Edit: Clearly this guess should have been $c_n + \sum_{i=1}^M k_{(i)}^n p_{(i) n}$, thanks Karene. Further, I was already aware of the issue whereby stability gives contradictions to e.g. $1+1+\cdots$ so my conjecture should also have reflected that.)
Given a specific, arbitrary series not of this form, can one actually/necessarily either (a) find two stable, linear, regular summation schemes for which the sum is defined, but for which we get distinct results; or (b) prove that no such scheme can define a value for the sum? (Motivation: pure curiosity.)

REPLY [5 votes]: I will assume that linearity is $$\sum(\alpha a_n+\beta b_n)=a\sum a_n+\beta\sum b_n.$$
If not, the constant coefficients below take them to be rational.
Denote by $S(a_1,a_2,...):=(0,a_1,a_2,...)$ the unilateral shift operator. Denote by $A$ the summation functional, that takes a series and returns its sum. We assume that $A$ is stable, linear, and regular.
We have that $A\circ S=A$, from stability. Moreover, stability is equivalent (assuming also linearity) to 

$$A\circ S=A.$$ 

In fact, if $A\circ S=A$, and since $S(a_1,0,0,...)=a_1$,
$$\begin{align}A(a_1,a_2,...)&=A(a_1,0,0,...)+A(0,a_2,a_3,...)\\&=a_1+A(0,a_2,...)\\&=a_1+A(a_2,a_3,...)\end{align},$$
which is the axiom of stability.
If $A(\vec{a})$, with $\vec{a}:=(a^{(1)},a^{(2)},...)$ a vector of sequences, and $a^{(i)}:=(a_{i1},a_{i2},...)$, can be determined by the axioms it means that there is a finite number of applications of the axioms to the sequence and to some convergent series that gives you an equation from which the sum id determined. 
Denote by $L_C:=C_kS^k+C_{k-1}S^{k-1}+...+C_1S+C_0I$ to be a linear finite-difference operator with constant coefficients, where the $C_i$ are vectors and the $S$ is applied to vectors of sequences componentwise. Then we can translate the above statement on being determined by the axioms into: 
There exist finitely many vectors of convergent series $\sum f_n^{(1)},\sum f_n^{(2)},...,\sum f_n^{(r)}$, and linear operators $L_{C^{(1)}}, L_{C^{(3)}},...,L_{C^{(r)}}$, and $L_C$, such that
$$L_C(a)=L_{C^{(1)}}(f^{(1)})+L_{C^{(3)}}(f^{(2)})+...+L_{C^{(r)}}(f^{(r)}),$$
and $d:=\det[C_k|C_{k-1}|...|C_0]\neq0$.
This is what allows to apply $A$ in both sides and obtain $$dA(\vec{a})=A(L_{C^{(1)}}(f^{(1)})+L_{C^{(3)}}(f^{(2)})+...+L_{C^{(r)}}(f^{(r)})),$$
from where the sum is computed.
Clearly the sequence in the right-hand side is convergent. Therefore we get that $$L_C(\vec{a})=f$$
for some $f:=(f_1,f_2,...)$, with $\sum f_n$ convergent (usual sense).
This answers your first question (as I understand it) about what series' values are determined by the axioms: 

Answer: These solutions of linear recurrence equations with constant coefficients adding up to a non-zero number, and with independent term being the term of a convergent series (the usual convergence).

You second question about different summations satisfying the axioms and defining sums in incompatible ways seems to be answered in the positive in Wikipedia (as dani_s pointed out). Maybe later I (or someone can give an explicit example of that).
It remains though the following

Question: Consider the series that have sums determined by the axioms (those solutions of recurrences). Are the values determined by the axioms consistent? In other words, is it possible to have a series satisfying two of these recurrence equations that then determine for the series two different values for its generalized sum?