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.
Chasse, Linear Preservers and Entire Functions with Restricted Zero Loci, Univer- sity of Hawai’i, PhD dissertation (2011)
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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.