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.
The Laguerre polynomials preserve real-rootedness
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The linear transformation that sends $x^n$ to the n'th Laguerre polynomial preserves real-rootedness.
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Zeros of linear combinations of Laguerre polynomials
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.