Non-commutative multiple bi-orthogonal polynomials: formal approach and integrability
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909821553344512 |
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| author | Doliwa, Adam |
| author_facet | Doliwa, Adam |
| contents | We define the non-commutative multiple bi-orthogonal polynomial systems, which simultaneously generalize the concepts of multiple orthogonality, matrix orthogonal polynomials and of the bi-orthogonality. We present quasideterminantal expressions for such polynomial systems in terms of formal bi-moments. The normalization functions for such monic polynomials satisfy the non-commutative Hirota equations, while the polynomials provide solution of the corresponding linear system. This shows, in particular, that our polynomial systems form a part of the theory of integrable systems. We study also a specialization of the problem to non-commutative multiple orthogonal polynomials, what results in the corresponding Hankel-type quasideterminantal expressions in terms of the moments. Moreover, such a reduction allows to introduce in a standard way the discrete-time variable and gives rise to an integrable system which is non-commutative version of the multidimensional discrete-time Toda equations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_02207 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Non-commutative multiple bi-orthogonal polynomials: formal approach and integrability Doliwa, Adam Exactly Solvable and Integrable Systems Numerical Analysis Mathematical Physics We define the non-commutative multiple bi-orthogonal polynomial systems, which simultaneously generalize the concepts of multiple orthogonality, matrix orthogonal polynomials and of the bi-orthogonality. We present quasideterminantal expressions for such polynomial systems in terms of formal bi-moments. The normalization functions for such monic polynomials satisfy the non-commutative Hirota equations, while the polynomials provide solution of the corresponding linear system. This shows, in particular, that our polynomial systems form a part of the theory of integrable systems. We study also a specialization of the problem to non-commutative multiple orthogonal polynomials, what results in the corresponding Hankel-type quasideterminantal expressions in terms of the moments. Moreover, such a reduction allows to introduce in a standard way the discrete-time variable and gives rise to an integrable system which is non-commutative version of the multidimensional discrete-time Toda equations. |
| title | Non-commutative multiple bi-orthogonal polynomials: formal approach and integrability |
| topic | Exactly Solvable and Integrable Systems Numerical Analysis Mathematical Physics |
| url | https://arxiv.org/abs/2510.02207 |