Spectral transformation associated with a perturbed $R_I$ type recurrence relation
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arXiv
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| Natura: | Preprint |
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2022
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| author | Shukla, Vinay Swaminathan, A. |
| author_facet | Shukla, Vinay Swaminathan, A. |
| contents | In this work, orthogonal polynomials satisfying $R_I$ type recurrence relation %$\mathcal{P}_{n+1}(z) = (z-c_n)\mathcal{P}_n(z)-λ_n (z-a_n)\mathcal{P}_{n-1}(z),$ with $\mathcal{P}_{-1}(z) = 0$ and $\mathcal{P}_0(z) = 1$ are analyzed when the recurrence coefficients are modified. The structural relationship between the perturbed and the unperturbed polynomials along with the spectral properties and spectral transformation of continued fraction are investigated. It is demonstrated that the transfer matrix method is computationally more efficient than the classical method for obtaining perturbed $R_I$ polynomials. Further, an interesting consequence of co-dilation on the Carathéodary function is presented. Finally, the study of co-recursion and co-dilation in connection to the unit circle is carried out with the help of an illustration. The interlacing and monotonicity of zeros between L-Jacobi polynomials and their perturbed forms are demonstrated. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2201_05422 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Spectral transformation associated with a perturbed $R_I$ type recurrence relation Shukla, Vinay Swaminathan, A. Classical Analysis and ODEs 42C05, 30B70, 30C15 In this work, orthogonal polynomials satisfying $R_I$ type recurrence relation %$\mathcal{P}_{n+1}(z) = (z-c_n)\mathcal{P}_n(z)-λ_n (z-a_n)\mathcal{P}_{n-1}(z),$ with $\mathcal{P}_{-1}(z) = 0$ and $\mathcal{P}_0(z) = 1$ are analyzed when the recurrence coefficients are modified. The structural relationship between the perturbed and the unperturbed polynomials along with the spectral properties and spectral transformation of continued fraction are investigated. It is demonstrated that the transfer matrix method is computationally more efficient than the classical method for obtaining perturbed $R_I$ polynomials. Further, an interesting consequence of co-dilation on the Carathéodary function is presented. Finally, the study of co-recursion and co-dilation in connection to the unit circle is carried out with the help of an illustration. The interlacing and monotonicity of zeros between L-Jacobi polynomials and their perturbed forms are demonstrated. |
| title | Spectral transformation associated with a perturbed $R_I$ type recurrence relation |
| topic | Classical Analysis and ODEs 42C05, 30B70, 30C15 |
| url | https://arxiv.org/abs/2201.05422 |