Related papers: Representation of hypergeometric products of highe…
In recent years, Karr's difference field theory has been extended to the so-called $R\Pi\Sigma$-extensions in which one can represent not only indefinite nested sums and products that can be expressed by transcendental ring extensions, but…
Recently, it has been shown constructively how a finite set of hypergeometric products, multibasic hypergeometric products or their mixed versions can be modeled properly in the setting of formal difference rings. Here special emphasis is…
We consider the following problem: Given a nested sum expression, find a sum representation such that the nested depth is minimal. We obtain a symbolic summation framework that solves this problem for sums defined, e.g., over…
A summation framework is developed that enhances Karr's difference field approach. It covers not only indefinite nested sums and products in terms of transcendental extensions, but it can treat, e.g., nested products defined over roots of…
We present an algebraic framework to represent indefinite nested sums over hypergeometric expressions in difference rings. In order to accomplish this task, parts of Karr's difference field theory have been extended to a ring theory in…
Using integration by parts relations, Feynman integrals can be represented in terms of coupled systems of differential equations. In the following we suppose that the unknown Feynman integrals can be given in power series representations,…
An algorithm for irreducible decomposition of representations of finite groups over fields of characteristic zero is described. The algorithm uses the fact that the decomposition induces a partition of the invariant inner product into a…
We present algorithms to solve coupled systems of linear differential equations, arising in the calculation of massive Feynman diagrams with local operator insertions at 3-loop order, which do {\it not} request special choices of bases.…
A general overview of the existing difference ring theory for symbolic summation is given. Special emphasis is put on the user interface: the translation and back translation of the corresponding representations within the term algebra and…
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…
Following up on a previous analysis of graph embeddings, we generalize and expand some results to the general setting of vector symbolic architectures (VSA) and hyperdimensional computing (HDC). Importantly, we explore the mathematical…
Take a multiplicative monoid of sequences in which the multiplication is given by Hadamard product. The set of linear combinations of interleaving monoid elements then yields a ring. For hypergeometric sequences, the resulting ring is a…
This paper describes algorithms to deal with nested symbolic sums over combinations of harmonic series, binomial coefficients and denominators. In addition it treats Mellin transforms and the inverse Mellin transformation for functions that…
An improved multi-summation approach is introduced and discussed that enables one to simultaneously handle indefinite nested sums and products in the setting of difference rings and holonomic sequences. Relevant mathematics is reviewed and…
We extend the symbolic representation to the ring of N=1 supersymmetric differential polynomials, and demonstrate that operations on the ring, such as the super derivative, Frechet derivative and super commutator, can be carried out in the…
We consider nested sums involving the Pochhammer symbol at infinity and rewrite them in terms of a small set of constants, such as powers of $\pi,$ $\log(2)$ or zeta values. In order to perform these simplifications, we view the series as…
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.…
To solve hard problems, AI relies on a variety of disciplines such as logic, probabilistic reasoning, machine learning and mathematical programming. Although it is widely accepted that solving real-world problems requires an integration…
In the present article the reduced integral representation of partitions in terms of harmonic products has been derived first by using hypergeometry and the new concept of fractional sum and secondly by studying the Fourier series of the…