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

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

2007-11-12

arxiv.org/abs/0711.1740v1

Mathematics

Classical Analysis and ODEs

11 pages

Scientific paper

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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