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Spectral Transformation Associated with a Perturbed \(R_I\) Type Recurrence Relation

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Abstract

In this work, orthogonal polynomials satisfying \(R_I\) type recurrence relation 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.

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Acknowledgements

The authors sincerely thank the anonymous referees for their constructive criticism, which helped to improve the manuscript. The first author would like to express his gratitude for the resources and support provided by Bennett University, Greater Noida, India, during the revision of this manuscript. This research work of the second author is supported by the Project No. CRG/2019/000200/MS of Science and Engineering Research Board, Department of Science and Technology, New Delhi, India.

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  1. Department of Mathematics, Indian Institute of Technology, Roorkee, Uttarakhand, 247667, India

    Vinay Shukla & A. Swaminathan

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  1. Vinay Shukla
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  2. A. Swaminathan
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Correspondence to A. Swaminathan.

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Communicated by Rosihan M. Ali.

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Shukla, V., Swaminathan, A. Spectral Transformation Associated with a Perturbed \(R_I\) Type Recurrence Relation. Bull. Malays. Math. Sci. Soc. 46, 169 (2023). https://doi.org/10.1007/s40840-023-01561-8

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