Zeros and asymptotics of a certain class of Sobolev-type Meixner polynomials

Abstract

UDC 517.587, 512.643 We consider a Sobolev-type inner product known from the literature. This inner product is a modification of the inner product of Meixner polynomials containing an additional term at a point $a\in\mathbb{R},$ involving the forward-difference operator $\Delta$ and a weight $\lambda>0.$ Here, the point $a$ is chosen to guarantee that the spectrum of  Meixner polynomials does not intersect the interval $(a, a+1).$ This inner product and its generalization already exist in the literature for various other measures. We  consider a specific measure. For the resulting orthonormal Sobolev-type polynomials, we obtain a five-term recurrence relation. The associated pentadiagonal matrix  is analyzed in relation to the Jacobi pencil matrix. The behavior of  zeros with respect to the parameters $a$ and $\lambda$ is investigated and the Plancherel–Rotach-type asymptotic behavior is obtained.