Analyses of multiplicity distributions by means of the Modi ed Negative Binomial Distribution and its KNO scaling function

We analyze various data of multiplicity distributions by means of the Modi ed Negative Binomial Distribution (MNBD) and its KNO scaling function, since this MNBD explains the oscillating behavior of the cumulant moment observed in ee annihilations, h-h collisions and e-p collisions. In the present analyses, we nd that the MNBD (discrete distributions) describes the data of charged particles in ee annihilations much better than the Negative Binomial Distribution (NBD). To investigate stochastic property of the MNBD, we derive the KNO scaling function from the discrete distribution by using a straightforward method and the Poisson transform. It is a new KNO function expressed by the Laguerre polynomials. In analyses of the data by using the KNO scaling function, we nd that the MNBD describes the data better than the gamma function. Thus, it can be said that the MNBD is one of useful formulas as well as NBD.