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Inequalities and monotonicity properties for gamma and q-gamma functions, pp. 309-323 in R. V. M. Zahar, ed., Approximation and Computation: A Festschrift in Honor of Walter Gautschi, ISNM, vol. 119, Birkhäuser, Boston-Basel-Berlin, 1994.Most of the errors in the original paper had to do with saying that certain functions related to the q-gamma function were not completely monotonic. We discovered these errors through reading the paper Some completely monotonic functions involving the q-gamma function, by Peng Gao, http://arxiv.org/abs/1011.3303.We also take the opportunity to correct some errors in other places including the statement and proof of Theorem 3.4.Abstract We prove some new results and unify the proofs of old ones involving complete monotonicity of expressions involving gamma and q-gamma functions, 0 < q < 1. Each of these results implies the infinite divisibility of a related probability measure. In a few cases, we are able to get simple monotonicity without having complete monotonicity. All of the results lead to inequalities for these functions. Many of these were motivated by the bounds in a 1959 paper by Walter Gautschi. We show that some of the bounds can be extended to complex arguments. 1 2 1−s , 0 < s < 1, x > 0. (1.4)

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We give two distinct approaches to finding bounds, as functions of the order i/, for the smallest real or purely imaginary zero of Bessel and some related functions. One approach is based on an old method due to Euler, Rayleigh, and others for evaluating the real zeros of the Bessel function Jv(x) when v > -1. Here, among other things, we extend this method to get bounds for the two purely imaginary zeros which arise in the case -2 < v < -1. If we use the notation j u i for the smallest positive zero, which approaches 0 as v -> -1 + , we can think of j^ as continued to-2 < v < -1, where it has negative values. We find an infinite sequence of successively improving upper and lower bounds for j^ in this interval. Some of the weakest, but simplest, lower bounds in this sequence are given by 4(z/ + 1) and 2 5 / 3 (i/ + l)[(i/ + 2)(u + 3)] 1 / 3 while a simple upper bound is 4(i/ + l)(v + 2) 1 / 2 . The second method is based on the representation of Bessel functions as limits of Lommel polynomials. In this case, the bounds for the zeros are roots of polynomials whose coefficients are functions of v. The earliest bounds found by this method already are quite sharp. Some are known in the literature though they are usually found by ad hoc methods. The same ideas are applied to get bounds for purely imaginary zeros of other functions such as J' v {x), J"(x), and OLJ V (X) + xJl(x).

ABSTRACT. 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 polynomials. We obtain new results on monotonicity of zeros of associated Laguerre, Al-Salam-Carlitz, Meixner and PoJlaczek polynomials. We also derive inequalities for the zeros of the Al-Salam-Carlitz and Meixner polynomials.

We improve previous results concerning the monotonicity in of f(?)x,, ,,: positive zero of the ultraspherical polynomial P,X)(x), and f(h) is a suitably chosen positive increasing _

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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 non-negative integer n, Hλ(t) has exactly n + 1 real zeros when n < λ ≤ n + 1, and that each zero increases from – ∞ to ∞ as λ increases. We establish a formula for the derivative of a zero with respect to the parameter λ; this derivative is a completely monotonic function of λ. By-products include some results on the regular sign behaviour of differences of zeros of Hermite polynomials as well as a proof of some inequalities, related to work of W. K. Hayman and E. L. Ortiz for the largest zero of Hλ(t). Similar results on zeros of certain confluent hypergeometric functions are given too. These specialize to results on the first, second, etc., positive zeros of Hermite polynomials.