Wilf–Zeilberger pairs are fundamental in the algorithmic theory of Wilf and Zeilberger for computer-generated proofs of combinatorial identities. Wilf–Zeilberger forms are their high-dimensional generalizations, which can be used for proving and discovering convergence acceleration formulas. This paper presents a structural description of all possible rational such forms, which can be viewed as an additive analog of the classical Ore–Sato theorem. Based on this analog, we show a structural decomposition of so-called multivariate hyperarithmetic expressions, which extend multivariate hypergeometric terms to the additive setting.

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

Although in theory we can decide whether a given D-finite function is transcendental, transcendence proofs remain a challenge in practice. Typically, transcendence is certified by checking certain incomplete sufficient conditions. In this paper we propose an additional such condition which catches some cases on which other tests fail.

In 2013, Pak and Panova proved the strict unimodality property of q-binomial coefficients (as polynomials in q) based on the combinatorics of Young tableaux and the semigroup property of Kronecker coefficients. They showed it to be true for all ℓ, m ≥ 8 and a few other cases. We propose a different approach to this problem based on computer algebra, where we establish a closed form for the coefficients of these polynomials and then use cylindrical algebraic decomposition to identify exactly the range of coefficients where strict unimodality holds. This strategy allows us to tackle generalizations of the problem, e.g., to show unimodality with larger gaps or unimodality of related sequences. In particular, we present proofs of two additional cases of a conjecture by Stanley and Zanello.

We identify tessellation-filtering ReLU neural networks that, when composed with another ReLU network, keep its non-redundant tessellation unchanged or reduce it.The additional network complexity modifies the shape of the decision surface without increasing the number of linear regions. We provide a mathematical understanding of the related additional expressiveness by means of a novel measure of shape complexity by counting deviations from convexity which results in a Boolean algebraic characterization of this special class. A local representation theorem gives rise to novel approaches for pruning and decision surface analysis.

Reconstructing a hypothetical recurrence equation from the first terms of an infinite sequence is a classical and well-known technique in experimental mathematics. We propose a variation of this technique which can succeed with fewer input terms.

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

We study zero-forcing (ZF) detection for multiple input/multiple output (MIMO) spatial multiplexing under transmit-correlated Rician fading for an NR× NTchannel matrix with rank-1 line-of-sight component. By using matrix transformations and multivariate statistics, our exact analysis yields the signal-to-noise ratio moment generating function (M.G.F.) as an infinite series of gamma distribution M.G.F.'s and analogous series for ZF performance measures, e.g., outage probability and ergodic capacity. However, their numerical convergence is inherently problematic with increasing Rician K-factor, NR, and NT. We circumvent this limitation as follows. First, we derive differential equations satisfied by the performance measures with a novel automated approach employing a computer-algebra tool that implements Gröbner basis computation and creative telescoping. These differential equations are then solved with the holonomic gradient method (HGM) from initial conditions computed with the infinite series. We demonstrate that HGM yields more reliable performance evaluation than by infinite series alone and more expeditious than by simulation, for realistic values of K, and even for NRand NTrelevant to large MIMO systems. We envision extending the proposed approaches for exact analysis and reliable evaluation to more general Rician fading and other transceiver methods.

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

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 is relevant in heterogeneous networks, has yielded for the ZF outage probability and ergodic capacity infinite-series expressions. Because they arose from expanding the confluent hypergeometric function1F1(·, ·, σ) around 0, they do not converge numerically at realistically high Rician K-factor values. Therefore, herein, we seek to take advantage of the fact that1F1(·, ·, σ) satisfies a differential equation, i.e., it is a holonomic function. Holonomic functions can be computed by the holonomic gradient method (HGM), i.e., by numerically solving the satisfied differential equation. Thus, we first reveal that the moment generating function (m.g.f.) and probability density function (p.d.f.) of the ZF signal-to-noise ratio (SNR) are holonomic. Then, from the differential equation for1F1(·, ·, σ), we deduce those satisfied by the SNR m.g.f. and p.d.f. and demonstrate that the HGM helps compute the p.d.f. accurately at practically relevant values of K. Finally, numerical integration of the SNR p.d.f. produced by HGM yields accurate ZF outage probability and ergodic capacity results.

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

A sequence fn(q) is q-holonomic if it satisfies a nontrivial linear recurrence with coefficients polynomials in q and qn. Our main theorems state that q-holonomicity is preserved under twisting, i.e., replacing q by ωq where ω is a complex root of unity, and under the substitution q → qα where α is a rational number. Our proofs are constructive, work in the multivariate setting of ∂-finite sequences and are implemented in the Mathematica package HolonomicFunctions. Our results are illustrated by twisting natural q-holonomic sequences which appear in quantum topology, namely the colored Jones polynomial of pretzel knots and twist knots. The recurrence of the twisted colored Jones polynomial can be used to compute the asymptotics of the Kashaev invariant of a knot at an arbitrary complex root of unity.

A variation of Zeilberger's holonomic ansatz for symbolic determinant evaluations is proposed which is tailored to deal with Pfaffians. The method is also applicable to determinants of skew-symmetric matrices, for which the original approach does not work. As Zeilberger's approach is based on the Laplace expansion (cofactor expansion) of the determinant, we derive our approach from the cofactor expansion of the Pfaffian. To demonstrate the power of our method, we prove, using computer algebra algorithms, some conjectures proposed in the paper "Pfaffian decomposition and a Pfaffian analogue of q-Catalan Hankel determinants" by Ishikawa, Tagawa, and Zeng. A minor summation formula related to partitions and Motzkin paths follows as a corollary.