Related papers: Parameterized Telescoping Proves Algebraic Indepen…
Telescopes are much more expensive than astronomers, so it is essential to minimize required sample sizes by using the most data-efficient statistical methods possible. However, the most commonly used model-independent techniques for…
In recent years, Z.-W. Sun proposed several sophisticated conjectures on congruences for finite sums with terms involving combinatorial sequences such as central trinomial coefficients, Domb numbers and Franel numbers. These sums are double…
Telescoping sums very naturally lead to probability distributions on ${\mathbb Z}^+$. But are these distributions typically cosmetic and devoid of motivation? In this paper we give three examples of "first occurrence" distributions, each…
Leveraging a general framework adapted from symbolic integration, a unified reduction-based algorithm for computing telescopers of minimal order for hypergeometric and q-hypergeometric terms has been recently developed. In this paper, we…
We present a new algorithm to compute minimal telescopers for rational functions in two discrete variables. As with recent reduction-based approaches, our algorithm has the important feature that the computation of a telescoper is…
The paper studies algebraic independence of certain reciprocal sums of Fibonacci and Lucas sequences. Also more general binary recurrences are considered. The main tool is Mahler's method reducing the investigation of the algebraic…
A summation is a shift-invariant ${\rm R}$-module homomorphism from a submodule of ${\rm R}[[\sigma]]$ to ${\rm R}$ or another ring. [11] formalized a method for extending a summation to a larger domain by telescoping. In this paper, we…
The article addresses the problem whether indefinite double sums involving a generic sequence can be simplified in terms of indefinite single sums. Depending on the structure of the double sum, the proposed summation machinery may provide…
Recently, $R\Pi\Sigma^*$-extensions have been introduced which extend Karr's $\Pi\Sigma^*$-fields substantially: one can represent expressions not only in terms of transcendental sums and products, but one can work also with products over…
Continuing a series of articles in the past few years on creative telescoping using reductions, we adapt Trager's Hermite reduction for algebraic functions to fuchsian D-finite functions and develop a reduction-based creative telescoping…
Multiple binomial sums form a large class of multi-indexed sequences, closed under partial summation, which contains most of the sequences obtained by multiple summation of products of binomial coefficients and also all the sequences with…
A complete reduction on a difference field is a linear operator that enables one to decompose an element of the field as the sum of a summable part and a remainder such that the given element is summable if and only if the remainder is…
We outline basic principles of a new method that gives a conceptual reasoning for and, at the same time, proofs of (super)congruences for truncated sums of arithmetic hypergeometric evaluations.
Creative telescoping applied to a bivariate proper hypergeometric term produces linear recurrence operators with polynomial coefficients, called telescopers. We provide bounds for the degrees of the polynomials appearing in these operators.…
An expression in terms of (cyclotomic) harmonic sums can be simplified by the quasi-shuffle algebra in terms of the so-called basis sums. By construction, these sums are algebraically independent within the quasi-shuffle algebra. In this…
In terms of the telescoping method, a simple binomial sum is given. By applying the derivative operators to the equation just mentioned, we establish several general harmonic number identities including some known results.
We consider a class of sums over products of Z-sums whose arguments differ by a symbolic integer. Such sums appear, for instance, in the expansion of Gauss hypergeometric functions around integer indices that depend on a symbolic parameter.…
An algebraic framework in which to study infinite sums is proposed, complementing and augmenting the usual topological tools. The framework subsumes numerous examples in the literature. It is developed using many varied examples, with a…
We present a general framework in the setting of difference ring extensions that enables one to find improved representations of indefinite nested sums such that the arising denominators within the summands have reduced degrees. The…
The Abramov-Petkovsek reduction computes an additive decomposition of a hypergeometric term, which extends the functionality of the Gosper algorithm for indefinite hypergeometric summation. We modify the Abramov-Petkovsek reduction so as to…