[论文解读] Hypergeometric Multiple Orthogonal Polynomials and Random Walks

Amílcar Branquinho, Juan E. Fernández-Díaz+2 more|arXiv (Cornell University)|Jul 1, 2021

Mathematical functions and polynomials参考文献 23被引用 6

一句话总结

本文通过高斯–博雷尔分解框架引入了超几何多重正交多项式，利用结构化矩阵系统 S 和 ~S 构造了 I 型与 II 型多重正交多项式。关键贡献在于推导出双正交关系 ∫Δ B(m)(x)Q(k)(x)dμ(x) = δm,k，建立了多项式序列与其相关线性形式之间的完整对偶性。

ABSTRACT

The recently found hypergeometric multiple orthogonal polynomials on the step-line by Lima and Loureiro are shown to be random walk polynomials. It is proven that the corresponding Jacobi matrix and its transpose, which are nonnegative matrices and describe higher recurrence relations, can be normalized to two stochastic matrices, dual to each other. Using the Christoffel-Darboux formula on the step-line and the Poincaré theory for non-homogeneous recurrence relations it is proven that both stochastic matrices are related by transposition in the large $n$ limit. These random walks are beyond birth and death, as they describe a chain in where transitions to the two previous states are allowed, or in the dual to the two next states.The corresponding Karlin-McGregor representation formula is given for these new Markov chains. The regions of hypergeometric parameters where the Markov chains are recurrent or transient are given. Stochastic factorizations, in terms of pure birth and of pure death factors, for the corresponding Markov matrices of types I and II, are provided.Twelve uniform Jacobi matrices and the corresponding random walks, related to a Jacobi matrix of Toeplitz type, and theirs stochastic or semi-stochastic matrices (with sinks and sources), that describe Markov chains beyond birth and death, are found and studied. One of these uniform stochastic cases, which is a recurrent random walk, is the only hypergeometric multiple random walk having a uniform stochastic factorization. The corresponding weights, Jacobi and Markov transition matrices and sequences of type II multiple orthogonal polynomials are provided. Chain of Christoffel transformations connecting the stochastic uniform tuples between them, and the semi-stochastic uniform tuples, between them, are presented.

研究动机与目标

• 通过 I 型与 II 型公式化方法系统构造多重正交多项式。
• 建立多项式序列与其对偶线性形式之间的双正交关系。
• 通过完美的高斯–博雷尔分解，将正交多项式理论推广至多重指标与向量多重指标。
• 通过正交性条件探索多重正交多项式与随机游走模型之间的联系。
• 在双权系统下，定义并分析以 ν = (m+1,m) 和 ν = (m+1,m+1) 索引的多重正交多项式结构。

提出的方法

• 利用高斯–博雷尔分解中的矩阵 S，构造 II 型多重正交多项式 B(m) = x^m + ∑_{i=0}^{m-1} S_{m,i}x^i。
• 通过对偶矩阵 ~S 和归一化常数 H_k，定义 I 型多重正交多项式 A(2m)_1, A(2m+1)_1, A(2m)_2, A(2m+1)_2。
• 引入线性形式 Q(m) = w₁A(m)_1 + w₂A(m)_2，以实现与 B(m) 的对偶性。
• 应用向量多重指标 ν(2m) = (m+1,m) 和 ν(2m+1) = (m+1,m+1)，定义 B(ν), A(ν)_1, 和 A(ν)_2，并满足特定次数约束。
• 建立 I 型正交性：当 deg ≤ ν₁−1, ν₂−1 且 j ≤ |ν|−2 时，有 ∫Δ x^j (A→ν,1 w₁ + A→ν,2 w₂) dμ = 0。
• 建立 II 型正交性：当 deg B→ν ≤ |ν| 且 j = 0,…,ν_a−1（a=1,2）时，有 ∫Δ B→ν(x) w_a(x) x^j dμ = 0。

实验结果

研究问题

• RQ1 如何从高斯–博雷尔分解框架中系统构造多重正交多项式？
• RQ2 I 型与 II 型多重正交多项式满足的精确正交性条件是什么？
• RQ3 线性形式 Q(m) 如何与双正交系统中的多项式序列 B(m) 相关联？
• RQ4 多重指标 ν(2m) = (m+1,m) 和 ν(2m+1) = (m+1,m+1) 在索引正交多项式中起到什么作用？
• RQ5 在何种条件下，双正交关系 ∫Δ B(m)(x)Q(k)(x)dμ(x) = δm,k 成立？

主要发现

• II 型多重正交多项式 B(m) 通过高斯–博雷尔分解中的矩阵 S 显式构造为 B(m) = x^m + ∑_{i=0}^{m-1} S_{m,i}x^i。
• I 型多重正交多项式 A(2m)_1, A(2m+1)_1, A(2m)_2, A(2m+1)_2 通过对偶矩阵 ~S 和归一化常数 H_k 定义。
• 线性形式 Q(m) = w₁A(m)_1 + w₂A(m)_2 被证明满足双正交关系 ∫Δ B(m)(x)Q(k)(x)dμ(x) = δm,k。
• 当 j = 0,…,|ν|−2 时，I 型正交性成立，且满足 deg A→ν,1 ≤ ν₁−1 与 deg A→ν,2 ≤ ν₂−1。
• 当 j = 0,…,ν_a−1（a=1,2）时，II 型正交性成立，且满足 deg B→ν ≤ |ν|。
• 在权重系统 (w₁,w₂,dμ) 具有完美高斯–博雷尔分解的假设下，该系统完全一致。