Related papers: Holonomic Tools for Basic Hypergeometric Functions

Symplectic integrators are the established standard for long-term simulations of nearly-integrable Hamiltonian systems due to their preservation of geometric structures. However, they suffer from an inherent limitation: secular phase-shift…

Motivated by applications in robotics and computer vision, we study problems related to spatial reasoning of a 3D environment using sublevel sets of polynomials. These include: tightly containing a cloud of points (e.g., representing an…

Chu and Zhang, in 2014, introduced hypergeometric transforms derived through the application of an Abel-type summation lemma to Dougall's ${}_{5}H_{5}$-series. These transforms were applied by Chu and Zhang to obtain accelerated rates of…

In this paper we introduce a new algebraic device, which enables us to treat the quaternions as though they were a commutative field. This is of interest both for its own sake, and because it can be applied to develop an "algebraic…

Guillera has introduced remarkable series expansions for $\frac{1}{\pi^2}$ of convergence rates $-\frac{1}{1024}$ and $-\frac{1}{4}$ via the Wilf-Zeilberger method. Through an acceleration method based on Zeilberger's algorithm and related…

The Boolean algebra of regular closed sets is prominent in topology, particularly as a dual for the Stone-Cech compactification. This algebra is also central for the theory of geometric computation, as a representation for combinatorial…

Before we proposed an algebraic technics for the Hamiltonian approach to the evolution systems of partial differential equations, including systems with constraints. Here we further develop this approach and present the defining system of…

Inclusions and extensions lie at the heart of physics and mathematics. The most relevant kind of inclusion in quantum systems is that of a von Neumann subalgebra, which is the focus of this work. We propose an object intrinsic to a given…

The continuous big $q$-Hermite polynomials are shown to realize a basis for a representation space of an extended $q$-oscillator algebra. An expansion formula is algebraically derived using this model.

This paper argues that automated proofs of identities for non-terminating hypergeometric series are feasible by a combination of Zeilberger's algorithm and asymptotic estimates. For two analogues of Saalsch\"utz' summation formula in the…

Geometric algebra (GA) is a mathematical tool for geometric computing, providing a framework that allows a unified and compact approach to geometric relations which in other mathematical systems are typically described using different more…

In this paper, we propose a general method to express explicitly the inversion and the connection coefficients between two basic hypergeometric polynomial sets. As application, we consider some $d$-orthogonal basic hypergeometric…

We present an approach to sums of random Hermitian matrices via the theory of spherical functions for the Gelfand pair $(\mathrm{U}(n) \ltimes \mathrm{Herm}(n), \mathrm{U}(n))$. It is inspired by a similar approach of Kieburg and K\"osters…

The ubiquity of the class of D-finite functions and P-recursive sequences in symbolic computation is widely recognized. In this thesis, the presented work consists of two parts related to this class. In the first part, we generalize the…

The concept of monogenic functions over real alternative $\ast$-algebras has recently been introduced to unify several classical monogenic (or regular) functions theories in hypercomplex analysis, including quaternionic, octonionic, and…

This paper improves the algorithms based on supporting halfspaces and quadratic programming for convex set intersection problems in our earlier paper in several directions. First, we give conditions so that much smaller quadratic programs…

In this paper, we focus on clarifying the concept of solving equations of degree greater than six using continuous functions or hypergeometric functions and providing another proof of the non-existence of algebraic solutions for equations…

The harmonic polylogarithms (hpl's) are introduced. They are a generalization of Nielsen's polylogarithms, satisfying a product algebra (the product of two hpl's is in turn a combination of hpl's) and forming a set closed under the…

Function approximation is a generic process in a variety of computational problems, from data interpolation to the solution of differential equations and inverse problems. In this work, a unified approach for such techniques is…

Real algebraic geometry is the study of semi-algebraic sets, subsets of $\R^k$ defined by Boolean combinations of polynomial equalities and inequalities. The focus of this thesis is to study quantitative results in real algebraic geometry,…