Article ID Journal Published Year Pages File Type
4606976 Journal of Approximation Theory 2015 10 Pages PDF
Abstract

We discuss the common zeros of the Laguerre polynomials Ln(α) and Ln−k(α+t), considering them as functions of tt. These common zeros are useful in discussing the interlacing of the zeros. Our main result is that if α≥0α≥0, and kk is a positive integer with 1≤k≤n−11≤k≤n−1, then for each tt in the interval 0≤t≤2k0≤t≤2k, excluding the values of tt for which Ln(α) and Ln−k(α+t) have a common zero, the zeros of these two polynomials interlace. Moreover, the interval 0≤t≤2k0≤t≤2k is the largest possible interval in which this interlacing property holds for all nn. We use the interlacing concept in an extended sense, originally due to Stieltjes.

Related Topics
Physical Sciences and Engineering Mathematics Analysis
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