Application of Gegenbauer Polynomials with Two Variables to Bi-univalency of Generalized Discrete Probability Distribution Via Zero-Truncated Poisson Distribution Series

Abstract

The present study is unique in exploring bi-univalent functions, which has recently garnered attention from many researchers in Geometric Function Theory (GFT). The uniqueness lies in utilizing a generalized discrete probability distribution and a zero-truncated Poisson distribution combined with generalized Gegenbauer polynomials featuring two variables. We aim to obtain coefficient bounds, the classical Fekete-Szegö inequality, and Hankel and Toeplitz determinants to generalize the probability of a gambler's ruin. Additionally, using the defined bi-univalent function classes contributes to the uniqueness of the obtained results.

Keywords

• bi-univalent function
• gegenbauer polynomials
• discrete probability
• hankel and toeplitz determinants
• zero-truncated-poisson series