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Rong-Hua Wang

Publications and source records attributed to Rong-Hua Wang.

At least 19 recordsLinked to original sources
arXiv2026

Polynomial reduction for $q$-holonomic sequences

This paper provides a (Laurent) polynomial reduction to $q$-holonomic sequences $F_k(q)$. We first characterize Laurent polynomials $\tilde{p}(x)$ such that the product $\tilde{p}(q^k)F_k(q)$ is summable. Then the reduction framework is given to decompose any given Laurent polynomial into a summable part and a remainder with lower degree. Finally, we introduce a power-partible reduction for $q$-holonomic sequences of which the recurrence relation satisfies a certain symmetry condition. The advantage is that it can not only simultaneously eliminate the highest-degree and lowest-degree terms of a Laurent polynomial satisfying a symmetry condition, but also guarantee the symmetry of the remainder. As applications, we apply the reduction to $q$-central-Delannoy numbers to derive new $q$-identities and $q$-congruences.

math.CO↗
arXiv2025

$q$-Congruences for Z.-W. Sun's generalized polynomials $w^{(α)}_k(x)$

In 2022, Z.-W. Sun defined \begin{equation*} w_k^{(α)}{(x)}=\sum_{j=1}^{k}w(k,j)^αx^{j-1}, \end{equation*} where $k,α$ are positive integers and $w(k,j)=\frac{1}{j}\binom{k-1}{j-1}\binom{k+j}{j-1}$. Let $(x)_{0}=1$ and $(x)_{n}=x(x+1)\cdots(x+n-1)$ for all $n\geq 1$. In this paper, it is proved by $q$-congruences that for any positive integers ${α,β, m,n,r}$, we have \begin{equation*} \frac{(2,n)}{n(n+1)(n+2)}\sum_{k=1}^{n}k^r(k+1)^r(2k+1)w_{k}^{(α)}(x)^{m}\in\mathbb{Z}[x], \end{equation*} \begin{equation*} \frac{(2,n)}{n(n+1)(n+2)}\sum_{k=1}^{n}(-1)^{k}k^r(k+1)^r(2k+1) w_{k}^{(α)}(x)^{m}\in\mathbb{Z}[x], \end{equation*} and \begin{equation*} \frac{2}{[n,n+1,\cdots,n+2β+1]}\sum_{k=1}^{n}(k)_β^r(k+β+1)_β^r(k+β) \prod_{i=0}^{2β-1}w_{k+i}^{(α)}(x)^m\in\mathbb{Z}[x], \end{equation*} where $[n,n+1,\cdots,n+2β+1]$ is the least common multiple of $n$, $n+1$, $\cdots$, $n+2β+1$. Taking $r=β=1$ above will confirm some of Z.-W. Sun's conjectures.

math.NT↗
arXiv2025

Congruences for sums of Delannoy numbers and polynomials

In this paper, we apply the power-partible reduction to study arithmetic properties of sums involving Delannoy numbers $D_k$ and polynomials $D_k(z)$. Let $v\in\bN$ and $p$ be an odd prime. It is proved that, for any $z\in\bZ\setminus\{0,-1\}$, there exist $c_v\in z^{-v}\bZ[z]$ and $\tilde{c}_v\in (z+1)^{-v}\bZ[z]$, both free of $p$ and can be determined mechanically, such that \begin{equation*} \sum_{k=0}^{p-1}(2k+1)^{2v}D_k(z)\equiv c_v \left(\frac{-z}{p}\right) \pmod {p} \end{equation*} if $\gcd(p,z)=1$ and \begin{equation*} \sum_{k=0}^{p-1}(-1)^k(2k+1)^{2v}D_k(z)\equiv \tilde{c}_v \left(\frac{z+1}{p}\right) \pmod {p} \end{equation*} if $\gcd(p,z+1)=1$. Here $(-)$ denotes the Legendre symbol. When $n$ is a power of $2$, we find there exist odd integers $ρ_v$ and even integers $\tildeρ_v$, both independent of $n$ and can be determined mechanically, such that \[ \sum_{k=0}^{n-1}(2k+1)^{2v+1}D_k\equiv ρ_v n \pmod {n^3} \] and \[ \sum_{k=0}^{n-1}(-1)^k(2k+1)^{2v+1}D_k\equiv \tildeρ_v n^2 \pmod {n^3}. \] The case $v=1$ in the last congruence confirms a conjecture of Guo and Zeng in 2012.

math.CO↗
arXiv2025

Non-minimality of minimal telescopers explained by residues

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.

cs.SC↗
arXiv2025

Arithmetic properties of generalized Delannoy polynomials and Schröder polynomials

Let $n$ be any nonnegative integer and \[ D_n^{(h)}(x)=\sum_{k=0}^{n}\binom{n+k}{2k}^{h}\binom{2k}{k}^{h}{x}^{k} \text{ and } S_{n}^{(h)}(x)=\sum_{k=0}^{n}\binom{n+k}{2k}^{h}C_{k}^{h}{x}^{k} \] be the generalized Delannoy polynomials and Schröder polynomials respectively. Here $C_k$ is the Catalan number and $h$ is a positive integer. In this paper, we prove that $$\begin{align*} & \frac{(2,n)}{n(n+1)(n+2)} \sum_{k=1}^{n}k^a(k+1)^a(2k+1)D_{k}^{(h)}(x)^{m}\in\mathbb{Z}[x],\\ &\frac{(2,hm-1,n)}{n(n+1)(n+2)} \sum_{k=1}^{n}(-1)^{k}k^a(k+1)^a(2k+1)D_{k}^{(h)}(x)^{m}\in\mathbb{Z}[x],\\ &\frac{(2,n)}{n(n+1)(n+2)} \sum_{k=1}^{n}k^a(k+1)^a(2k+1)S_{k}^{(h)}(x)^{m}\in\mathbb{Z}[x],\\ &\frac{(2,m-1,n)}{n(n+1)(n+2)} \sum_{k=1}^{n}(-1)^{k}k^a(k+1)^a(2k+1)S_{k}^{(h)}(x)^{m}\in\mathbb{Z}[x]. \end{align*}$$ Taking $a=1$ will confirm some of Z.-W. Sun's conjectures.

math.NT↗
arXiv2024

Power-Partible Reduction and Congruences for Apéry Numbers

In this paper, we introduce the power-partible reduction for holonomic (or, P-recursive) sequences and apply it to obtain a series of congruences for Apéry numbers $A_k$. In particular, we prove that, for any $r\in\mathbb{N}$, there exists an integer $\tilde{c}_r$ such that \begin{equation*} \sum_{k=0}^{p-1}(2k+1)^{2r+1}A_k\equiv \tilde{c}_r p \pmod {p^3} \end{equation*} holds for any prime $p>3$.

math.CO↗
arXiv2024

Rational reductions for holonomic sequences

Given a holonomic sequence $F(n)$, we characterize rational functions $r(n)$ so that $r(n)F(n)$ can be summable. We provide upper and lower bounds on the degree of the numerator of $r(k)$ and show the denominator of $r(n)$ can be read from annihilators of $F(k)$. This illustration provides the so-called rational reductions which can be used to generate new multi-sum equalities and congruences from known ones.

math.CO↗
arXiv2023

Power-partible Reduction and Congruences for Schröder Polynomials

In this note, we apply the power-partible reduction to show the following arithmetic properties of large Schröder polynomials $S_n(z)$ and little Schröder polynomials $s_n(z)$: for any odd prime $p$, nonnegative integer $r\in\mathbb{N}$, $\varepsilon\in\{-1,1\}$ and $z\in\mathbb{Z}$ with $\gcd(p,z(z+1))=1$, we have \[ \sum_{k=0}^{p-1}(2k+1)^{2r+1}\varepsilon^k S_k(z)\equiv 1\pmod {p}\quad \text{and} \quad \sum_{k=0}^{p-1}(2k+1)^{2r+1}\varepsilon^k s_k(z)\equiv 0\pmod {p}. \]

math.CO↗
arXiv2022

$q$-Rational Reduction and $q$-Analogues of Series for $π$

In this paper, we present a $q$-analogue of the polynomial reduction which was originally developed for hypergeometric terms. Using the $q$-Gosper representation, we describe the structure of rational functions that are summable when multiplied with a given $q$-hypergeometric term. The structure theorem enables us to generalize the $q$-polynomial reduction to the rational case, which can be used in the automatic proof and discovery of $q$-identities. As applications, several $q$-analogues of series for $π$ are presented.

math.CO↗
arXiv2022

Constructing minimal telescopers for rational functions in three discrete variables

We present a new algorithm for constructing minimal telescopers for rational functions in three discrete variables. This is the first discrete reduction-based algorithm that goes beyond the bivariate case. The termination of the algorithm is guaranteed by a known existence criterion of telescopers. Our approach has the important feature that it avoids the potentially costly computation of certificates. Computational experiments are also provided so as to illustrate the efficiency of our approach.

cs.SC↗
arXiv2022

Polynomial reduction for holonomic sequences and applications in $π$-series and congruences

Polynomial reduction, designed first for hypergeometric terms, can be used to automatically prove and generate new hypergeometric identities from old ones. In this paper, we extend the reduction method to holonomic sequences. As applications, we describe an algorithmic way to prove and generate new multi-summation identities. Especially we present new families of $π$-series involving Domb numbers and Franel numbers.

math.CO↗
arXiv2020

Congruences Related to Dual Sequences and Catalan Numbers

During the study of dual sequences, Sun introduced the polynomials \[ D_n(x,y)=\sum_{k=0}^{n}{n\choose k}{x\choose k}y^k\text{ and } S_n(x,y)=\sum_{k=0}^{n}\binom{n}{k}\binom{x}{k}\binom{-1-x}{k} y^k. \] Many related congruences have been established and conjectured by Sun. Here we generalize some of them by determining \[ \sum_{k=0}^{p-1}D_k(x_1,y_1)D_k(x_2,y_2)\pmod p \text{ and } \sum_{k=0}^{p-1}S_k(x_1,y_1)S_k(x_2,y_2)\pmod p \] for any odd prime $p$ and $p$-adic integers $x_i,\ y_i$ with $i\in\{1,2\}$. Considering the immediate connection between binomial coefficients and Catalan numbers, we also characterize \[ \sum_{n=0}^{p-1}\left(\sum_{k=0}^n {n \choose k} \frac{C_k}{a^k}\right)^2 \pmod {p}, \] where $C_k$ denotes the $k$th Catalan number, $a\in\mathbb{Z}\setminus \{0\}$ with $\gcd(a,p)=1$. These confirm and generalise some of Sun's conjectures.

math.CO↗
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.

cs.SC↗
arXiv2017

Lattice Walks in the Octant with Infinite Associated Groups

Continuing earlier investigations of restricted lattice walks in $\mathbb{N}^3$, we take a closer look at the models with infinite associated groups. We find that up to isomorphism, only 12 different infinite groups appear, and we establish a connection between the group of a model and the model being Hadamard.

math.CO↗
arXiv2016

Existence Problem of Telescopers: Beyond the Bivariate Case

In this paper, we solve the existence problem of telescopers for rational functions in three discrete variables. We reduce the problem to that of deciding the summability of bivariate rational functions, which has been solved recently. The existence criteria we present is needed for detecting the termination of Zeilberger's algorithm to the function classes studied in this paper.

cs.SC↗
arXiv2015

Infinite Orders and Non-$D$-finite Property of $3$-Dimensional Lattice Walks

Recently, Bostan and his coauthors investigated lattice walks restricted to the non-negative octant $\mathbb{N}^3$. For the $35548$ non-trivial models with at most six steps, they found that many models associated to a group of order at least $200$ and conjectured these groups were in fact infinite groups. In this paper, we first confirm these conjectures and then consider the non-$D$-finite property of the generating function for some of these models.

math.CO↗