Partial sums of analytic functions defined by binomial distribution and negative binomial distribution

The study of statistical distributions in a complex variable is one of the most vibrant areas of research. The complex analogue of many distributions has been studied. This article introduces the complex analogue of Binomial distribution and Negative Binomial distribution and studies certain geometrical properties of these analogues. It comprises the study of analytic functions defined by using the complex versions of Binomial distribution and Negative Binomial distribution functions. It includes the problems of finding the lower bounds of real parts of certain ratios of partial sums to the infinite series sums for these analytic functions. Such lower bounds are determined for the said analytic functions, for their first derivatives and for the Alexander transformation of these analytic functions.