Interlacing of zeros of Laguerre polynomials of equal and consecutive degree

Abstract

We investigate interlacing properties of zeros of Laguerre polynomials and where and . 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 and are interlacing for every and each is [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 and are interlacing for every and each is [Driver K, Muldoon ME. Common and interlacing zeros of families of Laguerre polynomials. J Approx Theory. 2015;193:89–98]. We derive conditions on and α, that determine the partial or full interlacing of the zeros of and the zeros of . We also prove that partial interlacing holds between the zeros of and when and . Numerical illustrations of interlacing and its breakdown are provided.