Accelerating indefinite hypergeometric summation algorithms

Abstract

Let K be a field of characteristic zero, x an independent variable, E the shift operator with respect to x, i.e., Ef ( x ) = f ( x + 1) for an arbitrary f ( x ). Recall that a nonzero expression F ( x ) is called a hypergeometric term over K if there exists a rational function r ( x ) ∈ K( x ) such that F ( x + 1)/ F ( x ) = r ( x ). Usually r ( x ) is called the rational certificate of F ( x ). The problem of indefinite hypergeometric summation (anti-differencing) is: given a hypergeometric term F ( x ), find a hypergeometric term G ( x ) which satisfies the first order linear difference equation ( E − 1) G ( x ) = F ( x ). (1) If found, write Σ x F ( x ) = G ( x ) + c , where c is an arbitrary constant.