Research Article Applications and Properties for Bivariate Bell-Based Frobenius- Type Eulerian Polynomials Waseem Ahmad Khan , 1 Maryam Salem Alatawi , 2 and Ugur Duran 3 1 Department of Mathematics and Natural Sciences, Prince Mohammad Bin Fahd University, P.O. Box 1664, Al Khobar 31952, Saudi Arabia 2 Department of Mathematics, Faculty of Science, University of Tabuk, Tabuk 71491, Saudi Arabia 3 Department of Basic Sciences of Engineering, Faculty of Engineering and Natural Sciences, İskenderun Technical University, Hatay 31200, Turkey Correspondence should be addressed to Waseem Ahmad Khan; wkhan1@pmu.edu.sa and Maryam Salem Alatawi; msoalatawi@ut.edu.sa Received 2 September 2022; Accepted 15 October 2022; Published 25 August 2023 Academic Editor: Sarfraz Nawaz Malik Copyright © 2023 Waseem Ahmad Khan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In this study, we introduce sine and cosine Bell-based Frobenius-type Eulerian polynomials, and by presenting several relations and applications, we analyze certain properties. Our first step is to obtain diverse relations and formulas that cover summation formulas, addition formulas, relations with earlier polynomials in the literature, and differentiation rules. Finally, after determining the first few zero values of the Eulerian polynomials, we draw graphical representations of these zero values. 1. Introduction In recent times, the use of sine and cosine polynomials has led to the definition and construction of generating func- tions for new families of special polynomials, such as Bernoulli, Euler, and Genocchi; see [1–4]. Fundamental properties and diverse applications for these polynomials have been provided by these types of studies. For instance, not only various implicit and explicit summation formulas, differentiation-integration formulas, symmetric identities, and a lot of relationships with the well-known polynomials have been deeply investigated but also graphical representa- tions of the zero values of these polynomials are drawn after determining them. Moreover, the aforementioned polyno- mials allow us to investigate worthwhile properties from a very basic procedure and assist to define novel types of spe- cial polynomials. Motivated by the above, in this paper, we define the cosine and sine Bell-based Frobenius-type Euler- ian polynomials and examine several properties and applica- tions. Our first step is to obtain diverse relations and formulas that cover summation formulas, addition formulas, relations with earlier polynomials in the literature, and dif- ferentiation rules. Finally, after determining the first few zero values of the Eulerian polynomials, we draw graphical repre- sentations of these zero values. Let ξ ∈ ℝ denotes the set of all real numbers and λ ∈ ℂ denotes the set of all complex numbers with λ ≠ 1. The Frobenius-type Eulerian polynomial of order α ∈ ℂ is intro- duced as follows (see [5–7]): 1 − λ e z λ−1 ð Þ − λ   α e ξz = 〠 ∞ j=0 A α ðÞ j ξjλ ð Þ z j j! , log λ λ − 1         > z jj: ð1Þ The Frobenius-type Eulerian polynomials have worked by many mathematicians; see [6–11]. Upon setting ξ =0, A ðαÞ j ðλÞ = A ðαÞ j ð0jλÞ are termed the Frobenius-type Eulerian numbers of order α. In view of (1), it can be readily observed that A α ðÞ j ξjλ ð Þ = 〠 j ν=0 j ν ! A α ðÞ ν λ ðÞξ j−ν , A α ðÞ j ξjλ ð Þ = λ − 1 ð Þ j ℍ α ðÞ j ξ λ − 1     λ   , ð2Þ Hindawi Journal of Function Spaces Volume 2023, Article ID 5205867, 10 pages https://doi.org/10.1155/2023/5205867