The Laguerre polynomials preserve real-rootedness

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keywords laguerrereal-rootednesslinearpolynomialpolynomialspreservepreservessends

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abstract

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
math.CA 2025-07 conditional novelty 7.0 of 10

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.