Indefinite sums of rational functions

Rational Normal Forms and Minimal Decompositions of Hypergeometric Terms

“…This formulation agrees with the minimal additive decomposition problem for rational functions (Abramov, 1975(Abramov, , 1995Pirastu and Strehl, 1995) because if T 2 ∈ K(x), then r = s = 1 and v is the denominator of T 2 .…”

“…(For definitions, see the last paragraph of this section.) First, recall the well-known decomposition problems for indefinite integrals (Hermite, 1872;Ostrogradsky, 1845) and indefinite sums (Abramov, 1975(Abramov, , 1995Paule, 1995;Pirastu and Strehl, 1995) of rational functions. Suppose for simplicity that a rational function R has no poles at non-negative arguments.…”

Shiftless decomposition and polynomial-time rational summation

“…Our rational summation algorithm in Section 5 is based on the criterion for rational summability in [3]. For clarity, the criterion was described there using the full factorization of the denominator of F inK[x], whereK is the algebraic closure of K. In fact, the criterion works also for a full factorization in K [x].…”

“…There were a number of algorithms and improvements developed over the following years, see for example [3,12,15,16,9]. In particular [16] gives a complete overview of these algorithms and improvements to them.…”

“…Using division with remainder to split off the polynomial part, which can be summed using, e.g., conversion to the falling factorial basis (see [6] §23.1), we may assume that deg f < deg g, so that F is proper. Let ρ be the dispersion of F , the maximal integer distance between roots of the denominator g. If ρ = 0 then we take R = 0 and H = F in (2), see [1,3,16]. In fact, the condition that the denominator of H has minimal degree is equivalent to the dispersion of H being zero.…”