Applications of q-Borel distribution series involving q-Gegenbauer polynomials to subclasses of bi-univalent functions

This study introduces a new class of bi-univalent functions in the open disk using q-Borel distribution series and q-Gegenbauer polynomials. It provides estimates for the Taylor coefficients |μ2| and |μ3| for this family of functions, as well as solutions for the Fekete-Szegö functional problems associated with this subclass. The study presents various innovative findings that result from the unique parameters used in the main results.


Introduction
In the year 1784, Legendre made the initial discovery of orthogonal polynomials (OP) [1].When particular model criteria are satisfied, ordinary differential equations are often solved by employing the operator (OP).In addition, the [2] notation plays a significant part in the field of approximation theory.
The polynomials Ξ  and Ξ  of order  and , respectively, are said to be orthogonal if where () is a well-defined function in the interval (, ), and as a result, the integral of all polynomials Ξ  () of finite order is well defined.
A type of (OP) is the Gegenbauer polynomial (GP).According to [3], symbolic connections exist between the generating mechanism of (GP) and the integral of the functions   , when conventional algebraic formulations are employed.This is the reason why many valuable inequalities have been discovered in the field of (GP).
The utilization of fractional calculus operators has been widely employed in the explanation and solution of issues in the field of applied sciences, as well as in Geometric Function, as documented in the source [4].The fractional -calculus is an extension of the standard fractional calculus.For further insights on the topic, it is recommended to consult a published source [5] and current literature, which might include references like [6][7][8].
The investigation of q-OP has yielded numerous significant discoveries and methodologies in the field of q-calculus.These encompass the q-analog of the binomial theorem, q-difference equations, and q-special functions.Moreover, the realm of q-OP has been employed to scrutinize q-integrals and q-series, which are indispensable tools in q-calculus.Notably, Quesne [9] recently proposed a reformulation of Jackson's q-exponential as a series of regular exponentials with well-defined coefficients.This breakthrough has significant implications for the theory of q-OP in this specific context and should be duly acknowledged.
The theory of (OP) is a subject that has received significant attention because of its wide range of applications in mathematics and physics.In recent years, there has been an increased utilization of (OP) and their analogues by researchers in the analysis of functions in the complex plane.This is particularly observed in the study of bi-univalent functions (see [10][11][12][13][14][15][16]).

Preliminaries
Consider the family A consisting of functions Φ of the form where  belongs to the complex unit disk Ω = { ∈ ℂ ∶ || < 1}, and Φ is analytic in Ω.Additionally,  must satisfy the normalization condition Φ ′ (0) = 0 and Φ(0) = 1.Furthermore, we denote by  a subclass of A that includes functions of Equation (2.1), which are also univalent in Ω.
The implementation of differential subordination of analytical functions has the potential to offer considerable benefits to the domain of (GFT).Miller and Mocanu [17] introduced the initial differential subordination problem, which has since been further investigated in [18].In their book [19], Miller and Mocanu provide a comprehensive overview of the developments in this field, including the corresponding publication dates.
If a discrete random variable  assumes values 1, 2, 3, … with corresponding probabilities, it is said to follow a Borel distribution respectively, where  is called the parameter.
Here, we introduce a power series that represents the probabilities linked with the -Borel distribution through its coefficients.
Now, we observe that the radius of convergence of the aforementioned power series is infinite, as can be deduced from the ratio test, given 0 ≤  ≤ 1.
Moreover, we define the series Next, we examine the convolution or Hadamard product, which defines a linear operator   () ∶ A ⟶ A on the function space A Askey and Ismail [34] developed a class of polynomials that can be regarded of as -analogues of the (GP) in 1983.Furthermore, the recurrence relations provided below can be utilized to interpret a specific set of polynomials, discovered by Chakrabarti et al. in 2006 [35], as -analogues of the (GP): Amourah et al. [36] and Alsoboh et al. [37] recently introduced subclasses of analytical and bi-univalent functions that use - (OP).In addition, Fekete-Szegö inequalities and constraints for the coefficients | are determined for functions belonging to these subclasses.Bi-univalent functions related to (OP) have recently been the subject of research by various writers, few to mention ( [38][39][40][41][42][43][44][45][46][47][48]).
The primary purpose of this paper is to initiate an investigation into the properties of bi-univalent functions associated to -(GP).

Definition and examples
The −(GP) is subordinate to some new bi-univalent function subclasses that we introduce in this section.
In light of the outcome established by Zaprawa [49], we investigate the ensuing Fekete-Szegö inequality that concerns functions belonging to the class  Σ (, ϝ; , ℸ; ).
Then, in view of (2.3), we conclude that )  .

Corollaries
Corollaries derived from Theorems 1 and 2 can be roughly associated with Examples 3.2, 3.3, and 3.4.

𝑞
) ) (5.1) (5.2) T. Al-Hawary, A. Alsoboh, A. Amourah et al. estimates for the Fekete-Szegö functional problems.We also explored additional results that were discovered through specialization of the parameters used in our primary results.In the future, it may be worthwhile to investigate the Hankel determinants of these classes.
We anticipate that the -defferintegral operator will have practical applications in various scientific domains, including technology and mathematics.

Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
2 Suppose  defined by equation (2.1) and belonging to the class Σ, is also a member of the class  Σ (, 1; , ℸ; ).