Martin E. Muldoon

Completely monotonic functions associated with the gamma function and its q-analogues

Several functions involving the gamma function Γ(x) and the q-gamma function Γq(x) are proved to be completely monotonic. Some of these results are used to establish the infinite divisibility of a number of probability distributions defined via their moment generating functions.

Inequalities and monotonicity properties for gamma and q -gamma functions

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.

Monotonicity of the Zeros of a Cross-Product of Bessel Functions

Our principal result is that for fixed

On the variation with respect to a parameter of zeros of bessel and q-bessel functions

\beta (0 0

A discrete approach to monotonicity of zeros of orthogonal polynomials

, the positive zeros of the cross-product \[J_{\nu + \beta } (x)K_\nu (\alpha x) - \alpha ^\beta J_\nu (x)K_{\nu + \beta } (\alpha x)\] increase with

Inequalities and Approximations for Zeros of Bessel Functions of Small Order

\nu

Inequalities and numerical bounds for zeros of ultraspherical polynomials

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Monotonicity and concavity properties of zeros of Bessel functions

{{ - \beta } / {2 \leqq \nu 1

Reciprocal Power Sums of Differences of Zeros of Special Functions

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Inequalities and monotonicity properties for zeros of Hermite functions

Abstract We apply what is essentially the Hellmann-Feynman theorem in a discrete setting to derive representations for the derivatives with respect to a parameter of the positive zeros of a family of entire functions. These representations prove that the zeros are monotone functions of the parameters involved. This family includes the Bessel functions Jv(x) and their basic analog Jv(2)(x; q). As a by-product we show that j v, 1 2 (v + 1) increases with v when v ϵ (−1, ∞), jv, 1 being the smallest positive zero of Jv(x). We also use the Lommel polynomials to derive improved lower bounds for the smallest positive zero of Jv(x).