In this paper, we get integral representations for the quintic Airy functions as the four linearly independent solutions of differential equation y(4) + xy = 0. Also, new integral representations for the products of these functions are obtained in terms of the Bessel functions and the Riesz fractional derivatives of these products are given.
We present a model in which scientists compete with each other in order to acquire status fortheir publications in a two-step-process: first, to get their work published in better journals, andsecond, to get this work cited in these journals. On the basis of two Maxwell-Boltzmann typedistribution functions of source publications we derive a distribution function of citingpublications over source publications. This distribution function corresponds very well to theempirical data. In contrast to all observations so far, we conclude that this distribution of citationsover publications, which is a crucial phenomenon in scientometrics, is not a power law, but amodified Bessel-function.
The exact probability density function for paired counting can be expressed in terms of modified Bessel functions of integral order when the expected blank count is known. Exact decision levels and detection limits can be computed in a straightforward manner. For many applications perturbing half-integer corrections to Gaussian distributions yields satisfactory results for decision levels. When there is concern about the uncertainty for the expected value of the blank count, a way to bound the errors of both types using confidence intervals for the expected blank count is discussed.
Authors:C. Donati-martin, M. Yor, and R. Ghomrasni
asymptotique des lois stables d'indice α, lorsque α tend vers 0, Prépublication n° 289, Laboratoire de Probabilités de l'Université Paris VI.
WATSON, G. N., A treatise on the theory of Besselfunctions , Cambridge