Certain new applications of Faber polynomial expansion for some new subclasses of $\upsilon$-fold symmetric bi-univalent functions associated with $q$-calculus

In this article, we define the $q$-difference operator and Salagean $q$-differential operator for $\upsilon$-fold symmetric functions in open unit disk $\mathcal{U}$ by first applying the concepts of $q$-calculus operator theory. Then, we considered these operators in order to construct new subclasses for $\upsilon$-fold symmetric bi-univalent functions. We establish the general coefficient bounds $|a_{\upsilon k+1}|$ for the functions in each of these newly specified subclasses using the Faber polynomial expansion method. Investigations are also performed on Feketo-Sezego problems and initial coefficient bounds for the function $h$ that belong to the newly discovered subclasses. To illustrate the relationship between the new and existing research, certain well-known corollaries of our main findings are also highlighted.