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Zeros of linear combinations of orthogonal polynomials
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abstract
Given a sequence of orthogonal polynomials $(p_n)_n$ with respect to a positive measure in the real line, we study the real zeros of finite combinations of $K+1$ consecutive orthogonal polynomials of the form $$ q_n(x)=\sum_{j=0}^K\gamma_jp_{n-j}(x),\quad n\ge K, $$ where $\gamma_j$, $j=0,\cdots ,K$, are real numbers with $\gamma_0=1$, $\gamma_K\not =0$ (which do not depend on $n$). We prove that for every positive measure $\mu$ there always exists a sequence of orthogonal polynomials with respect to $\mu$ such that all the zeros of the polynomial $q_n$ above are real and simple for $n\ge n_0$, where $n_0$ is a positive integer depending on $K$ and the $\gamma_j$'s.
Forward citations
Cited by 3 Pith papers
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An evolution of matrix-valued orthogonal polynomials
Connects scalar and matrix Gegenbauer polynomials through explicit expansions, unlocking new symmetries, generating functions, zero distributions, and differential-difference equations.
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Finite sums of consecutive Laguerre polynomials are real-rooted for large n exactly when an auxiliary coefficient polynomial has only real roots, with four Laguerre normalizations giving different counts of non-real zeros.
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Zeros of linear combinations of Hermite polynomials
Finite sums of consecutive Hermite polynomials have real-rootedness governed by a coefficient polynomial: real roots of P force real roots of the sum, and non-real roots of P appear one-for-one in high-degree sums.
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