Interlacing of zeros of Laguerre polynomials of equal and consecutive degree

We investigate interlacing properties of zeros of Laguerre polynomials L n ( alpha ) ( x ) and L n + 1 ( alpha + k ) ( x ) , alpha > - 1 , where n is an element of N and k is an element of { 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 L n ( alpha ) ( x ) and L n ( alpha + t ) ( x ) are interlacing for every n is an element of N and each alpha > - 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 L n ( alpha ) ( x ) and L n - 1 ( alpha + t ) ( x ) are interlacing for every n is an element of N and each alpha > - 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 is an element of N and alpha, alpha > - 1 that determine the partial or full interlacing of the zeros of L n ( alpha ) ( x ) and the zeros of L n ( alpha + 2 + k ) ( x ) , k is an element of { 1 , 2 } . We also prove that partial interlacing holds between the zeros of L n ( alpha ) ( x ) and L n - 1 ( alpha + 2 + k ) ( x ) when k is an element of { 1 , 2 } , n is an element of N and alpha > - 1 . Numerical illustrations of interlacing and its breakdown are provided.