Abstract
In the present work we deal with the quadratic decomposition of symmetric semiclassical polynomial sequences of class 2 orthogonal with respect to the positive definite weight \( | x^2-\frac{1}{2} |^p(1-x^2)^{-\frac{1}{2}}\), \( p > -1\), on \([-1,1]\). The coefficients of the three-term recurrence relation, the structure relation, the differential equation as well as some information about the zeros of the corresponding orthogonal polynomials are given. These results reduce to the Chebyshev case for \(p=0\).
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Acknowledgements
We would sincerely like to express special thanks to the referees for their interest and careful reading. Moreover, we are particularly indebted to them for suggesting to add either Proposition 4.2 and its corollary or the last section.
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Tounsi, M.I., Benabdallah, M. & Atia, M.J. A positive definite linear functional of class \(s=2\), generalization of Chebyshev polynomials. Period Math Hung 80, 195–210 (2020). https://doi.org/10.1007/s10998-019-00299-w
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DOI: https://doi.org/10.1007/s10998-019-00299-w