The variation of zeros of certain orthogonal polynomials

We show that the largest zero of a birth and death process polynomial increases (decreases) with a parameter ν if the birth rates and death rates are increasing (decreasing) functions of ν. A similar result is proved for the smallest zero of a birth and death process polynomial. These results are applicable to several sets of orthogonal polynomials. We show that the largest zero of a random walk polynomial is a monotone function of a parameter ν if certain coefficients related to the birth rates and the death rates are monotone functions of ν. We prove that if x_ν is a positive zero of a Lommel polynomial h_n,ν(x), ν > 0, then as ν increases x_ν will decrease but νx_ν will increase. Limiting cases of these results imply known facts concerning positive zeros of Bessel functions. We also establish similar results for a general class of discrete orthogonal polynomials.