application of gegenbauer polynomials with two variables to bi-univalency of generalized discrete probability distribution via zero-truncated poisson distribution series

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.