On a New Combination of Orthogonal Polynomials Sequences

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Home Editorial board Publication ethics Peer review guidelines Latest issue All issues Rules for authors Online submission system's guidelines Submit manuscript Contacts Address: Vatutina st. 53, Vladikavkaz, 362025, RNO-A, Russia Phone: (8672)23-00-54 E-mail: rio@smath.ru	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. Abstract: 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 + References 1. Chihara, T. S. An Introduction to Orthogonal Polynomials, New York, Gordon and Breach, 1978. 2. Maroni, P. Une Theorie Algebrique des Polynomes Orthogonaux. Application aux Polynomes Orthogonaux Semi-Classiques, Orthogonal Polynomials and their Applications, IMACS Annals on Computing and Applied Mathematics, eds. C. Brezinski et al., vol. 9, Basel, Baltzer, 1991, pp. 95-130. 3. Petronilho, J. On the Linear Functionals Associated to Linearly Related Sequences of Orthogonal Polynomials, Journal of Mathematical Analysis and Applications, 2006, vol. 315, no. 2, pp. 379-393. DOI: 10.1016/j.jmaa.2005.05.018. 4. Alfaro, M., Marcellan, F., Pena, A. and Rezola, M. L. On Linearly Related Orthogonal Polynomials and their Functionals, Journal of Mathematical Analysis and Applications, 2003, vol. 287, no. 1, pp. 307-319. DOI: 10.1016/S0022-247X(03)00565-1. 5. Alfaro, M., Marcellan, F., Pena, A. and Rezola, M. L. On Rational Transformations of Linear Functionals: Direct Problem, Journal of Mathematical Analysis and Applications, 2004, vol. 298, no. 1, pp. 171-183. DOI: 10.1016/j.jmaa.2004.04.065. 6. Alfaro, M., Marcellan, F., Pena, A. and Rezola, M. L. When Do Linear Combinations of Orthogonal Polynomials Yield New Sequences of Orthogonal Polynomials, Journal of Computational and Applied Mathematics, 2010, vol. 233, no. 6, pp. 1446-1452. DOI: 10.1016/j.cam.2009.02.060. 7. Alfaro, M., Pena, A., Rezola, M. L. and Marcellan, F. Orthogonal Polynomials Associated with an Inverse Quadratic Spectral Transform, Computers and Mathematics with Application, 2011, vol. 61, no. 4, pp. 888-900. DOI: 10.1016/j.camwa.2010.12.037. 8. Alfaro, M., Pena, A., Petronilho, J. and Rezola, M. L. Orthogonal Polynomials Generated by a Linear Structure Relation: Inverse Problem, Journal of Mathematical Analysis and Applications, 2013, vol. 401, no. 1, pp. 182-197. DOI: 10.1016/j.jmaa.2012.12.004. 9. Kwon, K. H., Lee, D. W., Marcellan, F. and Park, S. B. On Kernel Polynomials and Self-Perturbation of Orthogonal Polynomials, Annali di Matematica Pura ed Applicata, 2001, vol. 180, no. 2, pp. 127-146. DOI: 10.1007/s10231-001-8200-7. 10. Ronveaux, A. and Van Assche, W. Upward Extension of the Jacobi Matrix for Orthogonal Polynomials, Journal of Approximation Theory, 1996, vol. 86, no. 3, pp. 335-357. The sole copyright holder of the published work is the Founder of the Vladikavkaz Mathematical Journal. The terms of use of this work are governed by an open license (Creative Commons Attribution-NonCommercial 4.0 International). The use of metadata of the scientific article, including the title, abstract, author information, references, identifiers, and other bibliographic description elements for subsequent unrestricted use, is carried out under the terms of the CC BY or CC0 open licenses. ← Contents of issue


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