BESSEL POLYNOMIALS AND SOME CONNECTION FORMULAS IN TERMS OF THE ACTION OF LINEAR DIFFERENTIAL OPERATORS

Baghdadi Aloui     (University of Gabes, Higher Institute of Industrial Systems of Gabes Salah Eddine Elayoubi Str., 6033 Gabes, Tunisia)
Jihad Souissi     (University of Gabes, Faculty of Sciences of Gabes Erriadh Str., 6072 Gabes, Tunisia)

Abstract


In this paper, we introduce the concept of the \(\mathbb{B}_{\alpha}\)-classical orthogonal polynomials, where \(\mathbb{B}_{\alpha}\) is the raising operator \(\mathbb{B}_{\alpha}:=x^2 \cdot {d}/{dx}+\big(2(\alpha-1)x+1\big)\mathbb{I}\), with nonzero complex number \(\alpha\) and \(\mathbb{I}\) representing the identity operator. We show that the Bessel polynomials \(B^{(\alpha)}_n(x),\ n\geq0\), where \(\alpha\neq-{m}/{2}, \ m\geq -2, \ m\in \mathbb{Z}\), are the only \(\mathbb{B}_{\alpha}\)-classical orthogonal polynomials. As an application, we present some new formulas for polynomial solution.


Keywords


Classical orthogonal polynomials, Linear functionals, Bessel polynomials, Raising operators, Connection formulas

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References


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DOI: http://dx.doi.org/10.15826/umj.2022.2.001

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