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

摘要

We consider summation of consecutive values φ(v),φ(v + 1), ..., φ(w) of a meromorphic function φ(z) where v, w € ZZ. 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 ZZ is introduced.