In the present paper, a subclass of analytic and biunivalent functions by means of Gegenbauer polynomials is introduced. Certain coefficients bound for functions belonging to this subclass are obtained. Furthermore, the Fekete-Szegö problem for this subclass is solved. A number of known or new results are shown to follow upon specializing the parameters involved in our main results.

In the present analysis, we aim to construct a new subclass of analytic bi-univalent functions defined on symmetric domain by means of the Pascal distribution series and Gegenbauer polynomials. Thereafter, we provide estimates of Taylor–Maclaurin coefficients $\left|a_2\right|$ and $\left|a_3\right|$ for functions in the aforementioned class, and next, we solve the Fekete–Szegö functional problem. Moreover, some interesting findings for new subclasses of analytic bi-univalent functions will emerge by reducing the parameters in our main results.

Three subclasses of analytic and bi-univalent functions are introduced through the use of $q-$Gegenbauer polynomials, which are a generalization of Gegenbauer polynomials. For functions falling within these subclasses, coefficient bounds $\left|a_2\right|$ and $\left|a_3\right|$ as well as Fekete–Szegö inequalities are derived. Specializing the parameters used in our main results leads to a number of new results.

The Miller–Ross-type Poisson distribution is an important model for plenty of real-world applications. In the present analysis, we study and introduce a new class of bi-univalent functions defined by means of Gegenbauer polynomials with a Miller–Ross-type Poisson distribution series. For functions in each of these bi-univalent function classes, we have derived and explored estimates of the Taylor coefficients $\left|a_2\right|$ and $\left|a_3\right|$ and Fekete-Szegö functional problems for functions belonging to these new subclasses.

By using the generalization of the neutrosophic q-Poisson distribution series, we introduce a new subclass of analytic and bi-univalent functions defined in the open unit disk. We then apply the q-Gegenbauer polynomials to investigate the estimates for the Taylor coefficients and Fekete–Szegö type inequalities of the functions belonging to this new subclass. In addition, we consider several corollaries and the consequences of the results presented in this paper. The neutrosophic q-Poisson distribution is expected to be significant in a number of areas of mathematics, science, and technology.

The zero-truncated Poisson distribution is an important and appropriate model for many real-world applications. Here, we exploit the zero-truncated Poisson distribution probabilities to construct a new subclass of analytic bi-univalent functions involving Gegenbauer polynomials. For functions in the constructed class, we explore estimates of Taylor–Maclaurin coefficients $\left|a_2\right|$ and $\left|a_3\right|$, and next, we solve the Fekete–Szegő functional problem. A number of new interesting results are presented to follow upon specializing the parameters involved in our main results.