Confluent Hypergeometric Series Research Articles

A Two-Parameter Family of Hyper-Poisson Distributions

Abstract A two-parameter generalization of the Poisson distributions, called the “hyper-Poisson” family, is derived and discussed. It is in turn a special case of the three-parameter confluent hypergeometric series distributions. When one parameter is fixed, the hyper-Poisson distributions are also power series distributions. The distributions are classified as “sub-Poisson” or “super-Poisson” according as the variance is less than or greater than the mean. The latter category includes the truncated Poisson distributions. Certain properties of the hyper-Poisson distributions are considered, including moment, modified moment, and maximum likelihood estimators of the parameters. Estimation using moments is illustrated on Student's haemacytometer data.

A Two-Parameter Family of Hyper-Poisson Distributions

Abstract A two-parameter generalization of the Poisson distributions, called the “hyper-Poisson” family, is derived and discussed. It is in turn a special case of the three-parameter confluent hypergeometric series distributions. When one parameter is fixed, the hyper-Poisson distributions are also power series distributions. The distributions are classified as “sub-Poisson” or “super-Poisson” according as the variance is less than or greater than the mean. The latter category includes the truncated Poisson distributions. Certain properties of the hyper-Poisson distributions are considered, including moment, modified moment, and maximum likelihood estimators of the parameters. Estimation using moments is illustrated on Student's haemacytometer data.

Eigenfunctions following from Sommerfeld's Polynomial Method

The eigenfunctions following from Sommerfeld's polynomial method were determined in the general case. They are products of a convergence factor E and a polynomial P given by a hypergeometric series. The form of E depends upon whether P is expressible by the ordinary or the confluent hypergeometric series.