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Lixin Du

Publications and source records attributed to Lixin Du.

10 recordsLinked to original sources
arXiv2026

On the Summability Problem of Multivariate Rational Functions in the Mixed Case

Continuing previous work, this paper focuses on the summability problem of multivariate rational functions in the mixed case in which both shift and $q$-shift operators can appear. Our summability criteria rely on three ingredients including orbital decompositions, Sato's isotropy groups, and difference transformations. This work settles the rational case of the long-term project aimed at developing algorithms for symbolic summation of multivariate functions.

cs.SC↗
arXiv2025

Symmetric Division of Linear Ordinary Differential Operators

The symmetric product of two ordinary linear differential operators $L_1,L_2$ is an operator whose solution set contains the product $f_1f_2$ of any solution $f_1$ of $L_1$ and any solution $f_2$ of~$L_2$. It is well known how to compute the symmetric product of two given operators $L_1,L_2$. In this paper we consider the corresponding division problem: given a symmetric product $L$ and one of its factors, what can we say about the other factors?

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arXiv2025

Symbolic Summation of Multivariate Rational Functions

Symbolic summation as an active research topic of symbolic computation provides efficient algorithmic tools for evaluating and simplifying different types of sums arising from mathematics, computer science, physics and other areas. Most of existing algorithms in symbolic summation are mainly applicable to the problem with univariate inputs. A long-term project in symbolic computation is to develop theories, algorithms and software for the symbolic summation of multivariate functions. This paper will give complete solutions to two challenging problems in symbolic summation of multivariate rational functions, namely the rational summability problem and the existence problem of telescopers for multivariate rational functions. Our approach is based on the structure of Sato's isotropy groups of polynomials, which enables us to reduce the problems to testing the shift equivalence of polynomials. Our results provide a complete solution to the discrete analogue of Picard's problem on differential forms and can be used to detect the applicability of the Wilf-Zeilberger method to multivariate rational functions.

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arXiv2024

Solutions to the First Order Difference Equations in the Multivariate Difference Field

The bivariate difference field provides an algebraic framework for a sequence satisfying a recurrence of order two. Based on this, we focus on sequences satisfying a recurrence of higher order, and consider the multivariate difference field, in which the summation problem could be transformed into solving the first order difference equations. We then show a criterion for deciding whether the difference equation has a rational solution and present an algorithm for computing one rational solution of such a difference equation, if it exists. Moreover we get the rational solution set of such an equation.

math.CO↗
arXiv2023

On the Existence of Telescopers for P-recursive Sequences

We extend the criterion on the existence of telescopers for hypergeometric terms to the case of P-recursive sequences. This criterion is based on the concept of integral bases and the generalized Abramov-Petkovsek reduction for P-recursive sequences.

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arXiv2023

Hermite Reduction for D-finite Functions via Integral Bases

Trager's Hermite reduction solves the integration problem for algebraic functions via integral bases. A generalization of this algorithm to D-finite functions has so far been limited to the Fuchsian case. In the present paper, we remove this restriction and propose a reduction algorithm based on integral bases that is applicable to arbitrary D-finite functions.

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arXiv2021

Lazy Hermite Reduction and Creative Telescoping for Algebraic Functions

Bronstein's lazy Hermite reduction is a symbolic integration technique that reduces algebraic functions to integrands with only simple poles without the prior computation of an integral basis. We sharpen the lazy Hermite reduction by combining it with the polynomial reduction to solve the decomposition problem of algebraic functions. The sharpened reduction is then used to design a reduction-based telescoping algorithm for algebraic functions in two variables.

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arXiv2020

On the Existence of Telescopers for Rational Functions in Three Variables

Zeilberger's method of creative telescoping is crucial for the computer-generated proofs of combinatorial and special-function identities. Telescopers are linear differential or ($q$-)recurrence operators computed by algorithms for creative telescoping. For a given class of inputs, when telescopers exist and how to construct telescopers efficiently if they exist are two fundamental problems related to creative telescoping. In this paper, we solve the existence problem of telescopers for rational functions in three variables including 18 cases. We reduce the existence problem from the trivariate case to the bivariate case and some related problems. The existence criteria given in this paper enable us to determine the termination of algorithms for creative telescoping with trivariate rational inputs.

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arXiv2020

Integral P-Recursive Sequences

In an earlier paper, the notion of integrality known from algebraic number fields and fields of algebraic functions has been extended to D-finite functions. The aim of the present paper is to extend the notion to the case of P-recursive sequences. In order to do so, we formulate a general algorithm for finding all integral elements for valued vector spaces and then show that this algorithm includes not only the algebraic and the D-finite cases but also covers the case of P-recursive sequences.

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