The holonomic systems approach was proposed in the early 1990s by Doron Zeilberger. It laid a foundation for algorithmic treatment of function identities. Frédéric Chyzak later extended this framework introducing closely related notion ∂-finite functions and placing their manipulation on solid grounds. For practical purposes it is convenient to take advantage both concepts which not too much restriction: class that are contains many elementary (such as rational functions, algebraic...

We present a computer-aided, yet fully rigorous, proof of Ira Gessel's tantalizingly simply-stated conjecture that the number ways walking $2n$ steps in region $x+y \geq 0, y 0$ square-lattice with unit east, west, north, and south directions, start end at origin, equals $16^n\frac{(5/6)_n(1/2)_n}{(5/3)_n(2)_n}$ .

Elaborating on an approach recently proposed by Mark van Hoeij, we continue to investigate why creative telescoping occasionally fails find the minimal-order annihilating operator of a given definite sum or integral. We offer explanation based consideration residues.

We propose a version of the classical shape lemma for zero-dimensional ideals commutative multivariate polynomial ring to noncommutative setting in an algebra differential operators.

Designing mechanical devices, called linkages, that draw a given plane curve has been topic interested engineers and mathematicians for hundreds of years, recently also computer scientists. Already in 1876, Kempe proposed procedure solving the problem full generality, but his constructions tend to be extremely complicated. We provide novel algorithm produces much simpler works only parametric curves. Our approach is transform into factorization task over some noncommutative algebra. show how...

For multiple-input-multiple-output (MIMO) spatial-multiplexing transmission, zero-forcing (ZF) detection is appealing because of its low complexity. Our recent MIMO ZF performance analysis for Rician-Rayleigh fading, which relevant in heterogeneous networks, has yielded the outage probability and ergodic capacity infinite-series expressions. Because they arose from expanding confluent hypergeometric function <sub xmlns:mml="http://www.w3.org/1998/Math/MathML"...

Continuing a series of articles in the past few years on creative telescoping using reductions, we develop new algorithm to construct minimal telescopers for algebraic functions. This is based Trager's Hermite reduction and polynomial reduction, which was originally designed hyperexponential functions extended case this paper.

The conjecture that the orbit-counting generating function for totally symmetric plane partitions can be written as an explicit product formula, has been stated independently by George Andrews and David Robbins around 1983. We present a proof of this long-standing conjecture.

Our goal is to compute the minimal-order recurrence of colored Jones polynomial 7_4 knot, as well for first four double twist knots. As a corollary, we verify AJ Conjecture simplest knot with reducible non-abelian SL(2,C) character variety. To achieve our goal, use symbolic summation techniques Zeilberger's holonomic systems approach and an irreducibility criterion q-difference operators. For latter improved version qHyper algorithm Abramov-Paule-Petkovsek show that given operator has no...

We study the face-centered cubic (fcc) lattice in up to six dimensions. In particular, we are concerned with Green functions (LGFs) and return probabilities. Computer algebra techniques, such as method of creative telescoping, used for deriving an ODE a given LGF. For four- five-dimensional fcc lattices, give rigorous proofs ODEs that were conjectured by Guttmann Broadhurst. Additionally, find LGF six-dimensional lattice, result was not believed be achievable current computer hardware.

Laman graphs model planar frameworks that are rigid for a general choice of distances between the vertices. There finitely many ways, up to isometries, realize graph in plane. Such realizations can be seen as solutions systems quadratic equations prescribing pairs points. Using ideas from algebraic and tropical geometry, we provide recursive formula number complex such systems.

The Apagodu-Zeilberger algorithm can be used for computing annihilating operators definite sums over hypergeometric terms, or integrals hyperexponential functions. In this paper, we propose a generalization of which is applicable to arbitrary $\partial$-finite analogy the case, introduce notion proper We show that always succeeds these functions, and give tight priori bound order output operator.

We study q-holonomic sequences that arise as the colored Jones polynomial of knots in 3-space. The minimal-order recurrence for such a sequence is called (non-commutative) A-polynomial knot. Using "method guessing", we obtain this explicitly K_p = (-2, 3, 3+2p) pretzel p -5, ..., 5. This particularly interesting family since pairs (K_p, -K_{-p}) are geometrically similar (in particular, scissors congruent) with character varieties. Our computation non-commutative (a) complements done by...

We propose a differential analog of the notion integral closure algebraic function fields. present an algorithm for computing algebra defined by linear operator. Our is direct van Hoeij's bases

Computing the number of realizations a minimally rigid graph is notoriously difficult problem. Toward this goal, for graphs that are in plane, we take advantage recently published algorithm, which fastest available method, although its complexity still exponential. Combining computational results with theory constructing new by gluing, give lower bound on maximal possible (complex) given vertices. We extend these ideas to three dimensions and derive similar bounds, exploiting data from...

We prove an integrability criterion of order 3 for a homogeneous potential degree -1 in the plane. Still, this depends on some integer and it is impossible to apply directly except families potentials whose eigenvalues are bounded. To address issue, we use holonomic asymptotic computations with error control form V(r,\theta)=r^{-1} h(\exp(i\theta)) h polynomial less than 3. find then all meromorphically integrable form.