On the summation of rational functions

Parameterized Telescoping Proves Algebraic Independence of Sums

“…The summation criterion from [Abr71], [Pau95,Prop. 3.3] and its generalization to ΠΣ * -extensions are substantial:…”

“…Summarizing, parameterized telescoping in combination with ΠΣ * -fields gives a criterion to check algorithmically the transcendence of sums of type (1.3); see Theorem 5.1. Combining this criterion with results from summation theory, like [Abr71,Pau95,Abr03,Sch07a], shows that whole classes of sequences are transcendental. E.g., the harmonic numbers {H (i) n | i ≥ 1} with H (i) n := n k=1 1 k i are algebraically independent over Q(n).…”

Structural theorems for symbolic summation

“…Then, as predicted in Theorem 15, the solution (13) is given by a linear combination over Q in terms of the variables SumLeaf Q≤E (f ) = {s 1 , e, s 2,1,3 } plus one expression from Q(k, s 2 , s 3 , s 1,3 ).…”

“…This framework and extensions [42,43,44,23,48,24,45,25] generalize, e.g., the (q-)hypergeometric algorithms presented in [1,18,54,34,32,35,33,5,20,3], they cover as special case the summation of (q-)harmonic sums [10,51,29,11] arising, e.g., in particle physics, and they can treat classes of multi-sums that are out of scope of, e.g., the holonomic approach [53,52,15,14]. Karr's algorithm can be considered as the discrete analogue of Risch's algorithm [36,37] for indefinite integration.…”

The Euler-Maclaurin Formula and Differential Operators of Infinite Order

“…Recently, there has been a surge of interest in problems of summation thanks to the development of symbolic algorithms of summation of rational functions in papes by S. A. Abramov [3] and S. P. Polyakov [4], who call these problems "the indefinite summation". In some cases, however, it is more appropriate to use a more general difference equation than (1).…”