Mixed type multiple orthogonal polynomials associated with the modified Bessel functions and products of two coupled random matrices

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• 作者：Lun Zhang ^{lunzhang@fudan.edu.cn}
• 关键词：Mixed type multiple orthogonal polynomials ; Modified Bessel functions ; Integral representations ; Limiting forms ; Recurrence relations ; Random matrices
• 刊名：Journal of Approximation Theory
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：213
• 期：Complete
• 页码：92-119
• 全文大小：401 K
• 卷排序：213

文摘

We consider mixed type multiple orthogonal polynomials associated with a system of weight functions consisting of two vectors. One vector is defined in terms of scaled modified Bessel function of the first kind IμIμ and Iμ+1Iμ+1, the other vector is defined in terms of scaled modified Bessel function of the second kind KνKν and Kν+1Kν+1. We show that the corresponding mixed type multiple orthogonal polynomials exist. For the special case that each multi-index is on or close to the diagonal, basic properties of the polynomials and their linear forms are investigated, which include explicit formulas, integral representations, differential properties, limiting forms and recurrence relations. It comes out that, for specified parameters, the linear forms of these mixed type multiple orthogonal polynomials can be interpreted as biorthogonal functions encountering in recent studies of products of two coupled random matrices. This particularly implies a Riemann–Hilbert characterization of the correlation kernel, which provides an alternative way for further asymptotic analysis.