When do linear combinations of orthogonal polynomials yield new sequences of orthogonal polynomials?
Let \(\{P_n\}_{n\geq0}\) be a sequence of monic orthogonal polynomials with respect to a linear functional \(u\). The authors tackle linear combinations with constant coefficients \(Q_n=P_n+a_1P_{n-1}+\dots +a_kP_{n-k}\), \(n>k\). The main result provides necessary and sufficient conditions for the sequence \(\{Q_n\}\) to be orthogonal with respect to a linear functional \(v\). The authors give a simple algorithm to compute a polynomial \(h_k\) for which \(u=h_k v\). In the case \(k=2\) a complete description of all families \(\{P_n\}_{n\geq0}\) is obtained such that \(Q_n\) are orthogonal polynomials. Examples of such families are given.
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