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arXiv · 0711.1740

When do linear combinations of orthogonal polynomials yield new sequences of orthogonal polynomials?

M. Alfaro·F. Marcellan·A. Pena·M. L. Rezola
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Abstract

Given $\{P_n \}$ a sequence of monic orthogonal polynomials, we analyze their linear combinations $\{Q_n \}$with constant coefficients and fixed length $k+1$. Necessary and sufficient conditions are given for the orthogonality of the monic sequence $\{Q_n \}$ as well as an interesting interpretation in terms of the Jacobi matrices associated with $\{P_n \}$ and $\{Q_n \}$. Moreover, in the case $k=2$, we characterize the families $\{P_n \}$ such that the corresponding polynomials $\{Q_n \}$ are also orthogonal.

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M. Alfaro, F. Marcellan, A. Pena, M. L. Rezola. 2007-11-12. When do linear combinations of orthogonal polynomials yield new sequences of orthogonal polynomials?. https://arxiv.org/abs/0711.1740

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