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DOI: 10.46698/a8091-7203-8279-c

On a New Combination of Orthogonal Polynomials Sequences

Ali Khelil, K. , Belkebir, A. , Bouras, M. C.
Vladikavkaz Mathematical Journal 2022. Vol. 24. Issue 3.
In this paper, we are interested in the following inverse problem. We assume that \(\{P_{n}\} _{n\geq 0}\) is a monic orthogonal polynomials sequence with respect to a quasi-definite linear functional \(u\) and we analyze the existence of a sequence of orthogonal polynomials \(\{ Q_{n}\} _{n\geq 0}\) such that we have a following decomposition \(Q_{n}(x)+r_{n}Q_{n-1}(x)=P_{n}(x)+s_{n}P_{n-1}(x)+t_{n}P_{n-2}(x) +v_{n}P_{n-3}( x)\), \(n\geq 0\), when \(v_{n}r_{n}\neq 0,\) for every \(n\geq 4.\) Moreover, we show that the orthogonality of the sequence \(\{Q_{n}\}_{n\geq 0}\) can be also characterized by the existence of sequences depending on the parameters \(r_{n}\), \(s_{n}\), \(t_{n}\), \(v_{n}\) and the recurrence coefficients which remain constants. Furthermore, we show that the relation between the corresponding linear functionals is \(k( x-c) u=( x^{3}+ax^{2}+bx+d) v\), where \(c, a, b, d\in \mathbb{C}\) and \(k\in \mathbb{C}\setminus \{0\}\). We also study some subcases in which the parameters \(r_{n},\) \(s_{n},\) \(t_{n}\) and \(v_{n}\) can be computed more easily. We end by giving an illustration for a special example of the above type relation.
Keywords: orthogonal polynomials, linear functionals, inverse problem, Chebyshev polynomials
Language: English Download the full text  
For citation: Ali Khelil, K., Belkebir, A. and Bouras, M. C. On a New Combination of Orthogonal Polynomials Sequences, Vladikavkaz Math. J., 2022, vol. 24, no. 3, pp. 5-20. DOI 10.46698/a8091-7203-8279-c
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