Common and interlacing zeros of families of Laguerre polynomials

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.