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	Dear authors! Submission of all materials is carried out only electronically through Online Submission System in personal account. 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. ← Contents of issue


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