Interlacing of zeros of Laguerre polynomials of equal and consecutive degree

Abstract

We investigate interlacing properties of zeros of Laguerre polynomials Ln(α)(x) and Ln+1(α+k)(x), α>−1, where n∈N and k∈{1,2}. We prove that, in general, the zeros of these polynomials interlace partially and not fully. The sharp t-interval within which the zeros of two equal degree Laguerre polynomials Ln(α)(x) and Ln(α+t)(x) are interlacing for every n∈N and each α>−1 is 0<t≤ 2, [Driver K, Muldoon ME. Sharp interval for interlacing of zeros of equal degree Laguerre polynomials. J Approx Theory, to appear.], and the sharp t-interval within which the zeros of two consecutive degree Laguerre polynomials Ln(α)(x) and Ln−1(α+t)(x) are interlacing for every n∈N and each α>−1 is 0≤t≤ 2, [Driver K, Muldoon ME. Common and interlacing zeros of families of Laguerre polynomials. J Approx Theory. 2015;193:89–98]. We derive conditions on n∈N and α, α>−1 that determine the partial or full interlacing of the zeros of Ln(α)(x) and the zeros of Ln(α+2+k)(x), k∈{1,2}. We also prove that partial interlacing holds between the zeros of Ln(α)(x) and Ln−1(α+2+k)(x) when k∈{1,2}, n∈N and α>−1. Numerical illustrations of interlacing and its breakdown are provided.

Locations

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