Abstract
The purpose of this paper is to give, through the second degree character, new characterizations of a part of the family of quasi-symmetric forms. In fact, thanks to the Stieltjes function and also the moments, we give necessary and sufficient conditions for a regular form to be at the same time of the second degree, quasi-symmetric and semiclassical one of class two. We focus our attention not only on the link between all these forms and the Jacobi 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 \({{\mathcal {T}}}={\mathcal J}\left( -1/2, -1/2\right) \). The paper concludes by explicitly giving their characteristic elements of the structure relation and of the second order differential equation, which leads to interesting electrostatic models.
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Acknowledgements
We would sincerely like to express special thanks to the referees for their interest and careful reading of the manuscript. Moreover, we are particularly indebted to them for suggesting to add Proposition 4.9 and the last section.
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Ben Salah, I., Khalfallah, M. A description via second degree character of a family of quasi-symmetric forms. Period Math Hung 85, 81–108 (2022). https://doi.org/10.1007/s10998-021-00420-y
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DOI: https://doi.org/10.1007/s10998-021-00420-y
Keywords
- Orthogonal polynomials
- Classical and semiclassical forms
- Stieltjes function
- Second degree forms
- Differential equations
- Electrostatic model