compute closed forms of indefinite sums of rational functions
Rational(f, k, options)
rational function in k
(optional) equation of the form failpoints=true or failpoints=false
The Rational(f, k) command computes a closed form of the indefinite sum of f with respect to k.
Rational functions are summed using Abramov's algorithm (see the References section). For the input rational function f⁡k, the algorithm computes two rational functions s⁡k and t⁡k such that f⁡k=s⁡k+1−s⁡k+t⁡k and the denominator of t⁡k has minimal degree with respect to k. The non-rational part, ∑kt⁡k, is then expressed in terms of the digamma and polygamma functions.
If the option failpoints=true (or just failpoints for short) is specified, then the command returns a pair g,p,q, where
g is the closed form of the indefinite sum of f w.r.t. k,
p is a list containing the integer poles of f, and
q is a list containing the poles of s and t that are not poles of f.
See SumTools[IndefiniteSum][Indefinite] for more detailed help.
The following expression is rationally summable.
f ≔ 1n2+5⁢n−1
g ≔ Rational⁡f,n
g ≔ −13⁢n−32+12⁢5−13⁢n−12+12⁢5−13⁢n+12+12⁢5
Check the telescoping equation:
A non-rationally summable example.
f ≔ 13−57⁢x+2⁢y+20⁢x2−18⁢x⁢y+10⁢y215+10⁢x−26⁢y−25⁢x2+10⁢x⁢y+8⁢y2
f ≔ 20⁢x2−18⁢x⁢y+10⁢y2−57⁢x+2⁢y+13−25⁢x2+10⁢x⁢y+8⁢y2+10⁢x−26⁢y+15
g ≔ Rational⁡f,x
g ≔ −45⁢x+−725⁢y+3425⁢Ψ⁡x−45⁢y+35+1725⁢y+35⁢Ψ⁡x+25⁢y−1
Compute the fail points.
f ≔ 1n−2n−3+1n−5
g,fp ≔ Rational⁡f,n,'failpoints'
g,fp ≔ −1n−5−1n−4+1n−3+1n−2+1n−1,0..0,3..3,5..5,1,2,4
Indeed, f is not defined at n=0,3,5, and g is not defined at n=1,2,4.
Abramov, S.A. "Indefinite sums of rational functions." Proceedings ISSAC'95, pp. 303-308. 1995.
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