Abstract
In this work, we consider orthogonal polynomials via cubic decomposition in the framework of the second degree semiclassical class. We give a complete description, using the formal Stieltjes function and the moments, of semiclassical linear forms of class two which are of second degree such that their corresponding orthogonal polynomials sequences \(\{W_n\}_{n\ge 0}\) are obtained via the cubic decomposition \(W_{3n}(x)=P_n(x^3),~n\ge 0\). We focus our attention, not only on the link between all these forms and the second degree classical forms \({{\mathcal {T}}}_{p, q}={{\mathcal {J}}}(p-1/2, q-1/2)\), \(\; p, q\in {\mathbb {Z}},~p+q\ge 0\), but also on their connection with the Tchebychev form of the first kind \({{\mathcal {T}}}={{\mathcal {J}}}\left( -1/2, -1/2\right) \).
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The author would like to thank the anonymous referee for the suggestions they made which greatly improved the paper.
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Khalfallah, M. A Description of Second Degree Semiclassical Forms of Class Two Arising Via Cubic Decomposition. Mediterr. J. Math. 19, 30 (2022). https://doi.org/10.1007/s00009-021-01948-6
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DOI: https://doi.org/10.1007/s00009-021-01948-6
Keywords
- Orthogonal polynomials
- classical and semiclassical forms
- Stieltjes function
- cubic decomposition
- second degree forms