Related papers: A Difference Ring Theory for Symbolic Summation
By employing certain extended classical summation theorems, several surprising \pi and other formulae are displayed.
The \emph{sum-product phenomenon} predicts that a finite set $A$ in a ring $R$ should have either a large sumset $A+A$ or large product set $A \cdot A$ unless it is in some sense "close" to a finite subring of $R$. This phenomenon has been…
A novel method of summation for power series is developed. The method is based on the self-similar approximation theory. The trick employed is in transforming, first, a series expansion into a product expansion and in applying the…
By means of the telescoping method, we establish two sum- mation formulas on sine function. As the special cases of them, several interesting series expansions for $1/\pi^m$ and $\pi^m$.
We present efficient methods for calculating linear recurrences of hypergeometric double sums and, more generally, of multiple sums. In particular, we supplement this approach with the algorithmic theory of contiguous relations, which…
The use of unitary invariant subspaces of a Hilbert space $\mathcal{H}$ is nowadays a recognized fact in the treatment of sampling problems. Indeed, shift-invariant subspaces of $L^2(\mathbb{R})$ and also periodic extensions of finite…
We develop a theoretical study of non-terminating hypergeometric summations with one free parameter. Composing various methods in complex and asymptotic analysis, geometry and arithmetic of certain transcendental curves and rational…
Given a truncated perturbation expansion of a physical quantity, one can, under certain circumstances, obtain lower or upper bounds (or both) to the sum of the full perturbation series by using the Borel transform and a variational…
Diagrammatic reasoning (DR) is pervasive in human problem solving as a powerful adjunct to symbolic reasoning based on language-like representations. The research reported in this paper is a contribution to building a general purpose DR…
The development of mathematics has been characterized by the increasing interconnectivity of seemingly separate disciplines. Such interplay has been facilitated by a massive development in formalism; category theory has provided a common…
Solving symbolic reasoning problems that require compositionality and systematicity is considered one of the key ingredients of human intelligence. However, symbolic reasoning is still a great challenge for deep learning models, which often…
Expansion of higher transcendental functions in a small parameter are needed in many areas of science. For certain classes of functions this can be achieved by algebraic means. These algebraic tools are based on nested sums and can be…
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…
A new approach to summation of divergent field-theoretical series is suggested. It is based on the Borel transformation combined with a conformal mapping and does not imply the knowledge of the exact asymptotic parameters. The method is…
We give a definition of an integer-valued function $\sum_i \alpha_i x ^*_i$ derived from arrow diagrams for the ambient isotopy classes of oriented spherical curves. Then, we introduce certain elements of the free $\mathbb{Z}$-module…
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.…
Transseries expansions build upon ordinary power series methods by including additional basis elements such as exponentials and logarithms. Alternative summation methods can then be used to "resum" series to obtain more efficient…
Symbolic integration deals with the evaluation of integrals in closed form. We present an overview of Risch's algorithm including recent developments. The algorithms discussed are suited for both indefinite and definite integration. They…
By telescoping method, Sun gave some hypergeometric series whose sums are related to $\pi$ recently. We investigate these series from the point of view of Gosper's algorithm. Given a hypergeometric term $t_k$, we consider the Gosper…
The combination of the group ring setting with the methods of character theory allows an elegant and powerful analysis of various combinatorial structures, via their character sums. These combinatorial structures include difference sets,…