混合型Jacobi系统的Askey方案：第一类和第二类Laguerre极限及Hermite极限

中文摘要 AI 辅助

为混合型的Jacobi多重正交系统发展了一个Askey型汇合方案，该系统具有$q$个行权重和$p$个列权重。将Jacobi变量在$0$附近和$1$附近重新标度，分别产生第一类和第二类Laguerre系统。从第一类Laguerre、直接从第二类以及直接从Jacobi出发，得到一个具有一个Gaussian-gamma行和$q-1$个指数卷积的Hermite系统。给出了显式的极限权重和正交形式，以及其混合矩的收敛性。对于每个固定指标，在非消失条件下，递推系数和双对角因子的元素也在极限下收敛到第一类Laguerre。第二类Laguerre族由加性gamma卷积描述。其矩行列式在任意正实数形状参数和近对角行多重指标下成立一个封闭乘积公式，从而给出显式的正规性条件。其形式允许围道和Rodrigues表示，并具有终止超几何分量：$A$为多重Kampé de Fériet级数，当$q\ge2$时每个$B$分量至多为$q+1$个Srivastava-Daoust级数。对于Hermite族，在所述分离条件下，正规性在每个Laguerre可容许指标处成立，且这些指标构成一个无限集。其分量具有有限的Hermite展开和终止的Srivastava-Daoust表示。三条路径给出相同的极限矩和归一化分量，其中Jacobi路径是单次极限，另外两条是迭代极限。附录处理了几个导数行，其中连接行列式在有限指标范围之外消失，具有精确的秩损失和发散分量。单行约化恢复了经典的第二类多重Laguerre和Hermite系统。

英文摘要

An Askey-type confluence scheme is developed for Jacobi multiple orthogonal systems of mixed type, with $q$ row weights and $p$ column weights. Rescaling the Jacobi variable near $0$ and near $1$ produces Laguerre systems of the first and second kinds. A Hermite system with one Gaussian--gamma row and $q-1$ exponential convolutions is obtained from Laguerre of the first kind, directly from the second kind, and directly from Jacobi. Explicit limiting weights and orthogonal forms are given, together with convergence of their mixed moments. For each fixed index, the recurrence coefficients and the entries of the bidiagonal factors also converge in the limit to Laguerre of the first kind, under nonvanishing conditions. The Laguerre family of the second kind is described by additive gamma convolutions. A closed product formula for its moment determinants holds for arbitrary positive real shape parameters and near-diagonal row multi-indices, yielding explicit normality conditions. Its forms admit contour and Rodrigues representations with terminating hypergeometric components: a multiple Kampé de Fériet series for $A$, and at most $q+1$ Srivastava--Daoust series for each $B$-component when $q\ge2$. For the Hermite family, normality holds at every Laguerre-admissible index under the stated separation conditions, and these indices form an infinite set. Its components have finite Hermite expansions and terminating Srivastava--Daoust representations. The three routes give the same limiting moments and normalized components, the Jacobi one being a single limit and the other two iterated. An appendix treats several derivative rows, where the connection determinant vanishes beyond a finite index range, with exact rank loss and diverging components. The one-row reductions recover the classical multiple second-kind Laguerre and Hermite systems.

发表机构

• Complutense University of Madrid（马德里康普顿斯大学）

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