Analytic Solutions of Linear Difference Equations, Formal Series, and Bottom Summation

We consider summation of consecutive values ϕ(v), ϕ(v+1), ..., ϕ(w) of a meromorphic function ϕ(z) where v,w ∈ Z. We assume that ϕ(z) satisfies a linear difference equation L(y) = 0 with polynomial coefficients, and that a summing operator for L exists (such an operator can be found - if it exists - by the Accurate Summation algorithm, or alternatively, by Gosper's algorithm when ord L = 1). The notion of bottom summation which covers the case where ϕ(z) has poles in Z is introduced.