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関西可積分系セミナー (2000年11月10, 14, 21日)

日時
2000年11月10日(金)16時30分-18時, 11月14日(火)15時-16時30分, 11月21日(火)16時30分-18時
場所
大阪大学基礎工学研究科数理科学ディスプレイ室(J617)
Alexei Zhedanov (Donetsk Institute of Physics and Technology, Ukraine)
Orthogonal polynomials, biorthogonal rational functions, numerical algorithms, and integrable chains I, II, III

We consider general theory of spectral transformations for orthogonal, biorthogonal polynomials and biorthogonal rational functions. Such transformations are equivalent to multiplying (dividing) of the weight function by a polynomial function and to adding arbitrary discrete masses. For orthogonal polynomials all spectral transformations are reduced to a superposition of a finite number of Christoffel and Geronimus transforms. For biorthogonal Laurent polynomials and biorthogonal rational functions it is possible to construct similar spectral transformations. We show that the most general “classical” orthogonal polynomials (including Askey-Wilson and Askey-Ismail polynomials) are obtained under self-similarity condition for corresponding chain of spectral transformations. There are intimate relations between spectral transformations on the one hand and numerical algorithms on the other hand. In turn, many numerical algorithms are equivalent to different integrable chains with discrete time. We show how some well known integrable chains (like Toda and relativistic Toda chains) are related to orthogonal and biorthogonal polynomials. Moreover, we derive a new integrable R-II chain which leads to a new class of rational functions which are biorthogonal on elliptic grids.