Burchnall–Chaundy polynomials and the Laurent phenomenon
Burchnall–Chaundy polynomials and the Laurent phenomenon
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DOI：
10.1088/1751-8113/48/20/205201
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发表时间：
2014-07
期刊：
影响因子：
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通讯作者：
A. Veselov;R. Willox
中科院分区：
文献类型：
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作者：
A. Veselov;R. Willox
Burchnall-Chaundy多项式Pn（z）由微分递归关系Pn + 1 ′（z）Pn − 1（z）− Pn + 1（z）Pn − 1 ′（z）= Pn（z）2？其中P − 1（z）= P 0（z）= 1。?>这种递推关系的多项式通解并不明显，类似于Somos序列的可积性和Laurent现象。我们更详细地讨论这种并行，并将其推广到两个差分方程Q n + 1（z + 1）Q n − 1（z）-Q n + 1（z）Q n − 1（z + 1）= Q n（z）Q n（z + 1）？>且R n + 1（z + 1）R n − 1（z − 1）− R n + 1（z − 1）R n − 1（z + 1）= R n 2（z）？>与Hirota-Miwa和Dodgson八面体方程的两种不同的KdV型约化有关。作为推论，我们得到了关于初始数据Pn（0）的Burchnall-Chaundy多项式的一种新形式。，这表明是劳伦特。
The Burchnall–Chaundy polynomials Pn(z) are determined by the differential recurrence relation P n + 1 ′ ( z ) P n − 1 ( z ) − P n + 1 ( z ) P n − 1 ′ ( z ) = P n ( z ) 2 ?> with P − 1 ( z ) = P 0 ( z ) = 1 . ?> The fact that this recurrence relation has all solutions polynomial is not obvious and is similar to the integrality of Somos sequences and the Laurent phenomenon. We discuss this parallel in more detail and extend it to two difference equations Q n + 1 ( z + 1 ) Q n − 1 ( z ) − Q n + 1 ( z ) Q n − 1 ( z + 1 ) = Q n ( z ) Q n ( z + 1 ) ?> and R n + 1 ( z + 1 ) R n − 1 ( z − 1 ) − R n + 1 ( z − 1 ) R n − 1 ( z + 1 ) = R n 2 ( z ) ?> related to two different KdV-type reductions of the Hirota–Miwa and Dodgson octahedral equations. As a corollary we have a new form of the Burchnall–Chaundy polynomials in terms of the initial data P n ( 0 ) ?> , which is shown to be Laurent.