Exceptional Laurent biorthogonal polynomials through spectral transformations of generalized eigenvalue problems |
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Authors: | Yu Luo Satoshi Tsujimoto |
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Affiliation: | 1. Department of Mathematics, Zhejiang University of Technology, Hangzhou, China;2. Graduate School of Informatics, Kyoto University, Kyoto, Japan |
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Abstract: | A formulation is given for the spectral transformation of the generalized eigenvalue problem through the decomposition of the second-order differential operators. This allows us to construct some Laurent biorthogonal polynomial systems with gaps in the degree of the polynomial sequence. These correspond to an exceptional-type extension of the orthogonal polynomials, as an extension of the Laurent biorthogonal polynomials. Specifically, we construct the exceptional extension of the Hendriksen–van Rossum polynomials, which are biorthogonal analogs of the classical orthogonal polynomials. Similar to the cases of exceptional extensions of classical orthogonal polynomials, both state-deletion and state-addition occur. |
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Keywords: | differential operators exceptional Laurent biorthogonal polynomial generalized eigenvalue problem Hendriksen–Van Rossum polynomials Laurent biorthogonal polynomials |
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