We study the monotonicity with respect to a parameter of zeros of orthogonal polynomials. Our method uses the tridiagonal (Jacobi) matrices arising from the three-term recurrence relation for the… (More)

We present bounds and approximations for the smallest positive zero of the Laguerre polynomial L n (x) which are sharp as α → −1. We indicate the applicability of the results to more general… (More)

There is a contrast between the two sets of functional equations f0(x + y) = f0(x)f0(y) + f1(x)f1(y), f1(x + y) = f1(x)f0(y) + f0(x)f1(y), and f0(x − y) = f0(x)f0(y)− f1(x)f1(y), f1(x − y) =… (More)

We present a convergent asymptotic formula for the zeros of the Hermite functions as λ → ∞. It is based on an integral formula due to the authors for the derivative of such a zero with respect to λ.… (More)

Convexity properties are often useful in characterizing and nding bounds for special function and their zeros as well as in questions concerning the existence and uniqueness of zeros in certain… (More)

We give two distinct approaches to finding bounds, as functions of the order ν, for the smallest real or purely imaginary zero of Bessel and some related functions. One approach is based on an old… (More)

We study the variation of the zeros of the Hermite function Hλ(t) with respect to the positive real variable λ. We show that, for each nonnegative integer n, Hλ(t) has exactly n + 1 real zeros when n… (More)

For each m (= 1, . . . , n) the n-th Laguerre polynomial L n (x) has an m-fold zero at the origin when α = −m. As the real variable α → −m, it has m simple complex zeros which approach 0 in a… (More)

Suppose that the function q(t) in the diierential equation (1) y 00 + q(t)y = 0 is decreasing on (b;1) where b 0. We give conditions on q which ensure that (1) has a pair of solutions y 1 (t); y 2… (More)