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Thank you for your answer which reminds me to perform more experiments to explore properties. Do you mind to tell me what are the tools you use to find roots programmatically and also to visualize them? Thanks.
@RobertIsrael Sorry for the inaccuracy:) $k$ is an integer less than $n$, we may suppose $k=\beta n$ with $0<\beta<1$. By "largest", I mean greatest real root.
