 
Summary: On summation of Precursive sequences
S. A. Abramov
Russian Academy of Sciences
Dorodnicyn Computing Centre
Vavilova 40, 119991, Moscow GSP1, Russia
sabramov@ccas.ru
Abstract
We consider sequences which satisfy a linear recurrence equation
Ly = 0 with polynomial coe cients. A criterion, i.e., a neces
sary and su cient condition is proposed for validity of the discrete
NewtonLeibniz formula when a primitive (an inde nite sum) Rt of
a solution t of Ly = 0 is obtained either by Gosper's algorithm or
by the Accurate Summation algorithm (the operator R has rational
function coe cients, ordR = ordL ; 1 in the Gosper case ordL = 1,
ordR = 0). Additionally we show that if Gosper's algorithm succeeds
on L, ordL = 1, then Ly = 0 always has some nonzero solutions
t, de ned everywhere, such that the discrete NewtonLeibniz formulaPw
k=v t(k) = u(w+1);u(v) is valid for u = Rt and any integer bounds
v w.
1 Introduction
