J. Math. Computer Sci., 35 (2024), 291–303 Online: ISSN 2008-949X Journal Homepage: www.isr-publications.com/jmcs Properties and applications of Bell polynomials of two vari- ables Mohra Zayed a , Shahid Ahmad Wani b,* , William Ram´ırez c,e,* , Mohammad A. Alqudah d , Fabio Fuentes Gandara e a Mathematics Department, College of Science, King Khalid University, Abha 61413, Saudi Arabia. b Department of Applied Sciences, Symbiosis Institute of Technology, Symbiosis International (Deemed University) (SIU), Pune, Maharashtra, India. c Section of Mathematics, International Telematic University Uninettuno, Corso Vittorio Emanuele II, 39, 00186 Rome, Italy. d School of Basic Sciences and Humanities, German Jordanian University, Amman, 11180, Jordan. e Department of Natural and Exact Sciences, Universidad de la Costa, Calle 58 N 55-66, 080002 Barranquilla, Colombia. Abstract In this article, we introduce Bell polynomials of two variables within the framework of generating functions and explore various properties associated with them. Specifically, we delve into explicit representations, summation formulae, recurrence relations, and addition formulas. Additionally, we present the matrix form and product formula for these polynomials. Finally, we introduce the two-variable Bell-based Stirling polynomials of the second kind and outline their corresponding results. This study contributes to a deeper understanding of the properties and applications of Bell polynomials in mathematical analysis. Keywords: Special polynomials, monomiality principle, explicit form, operational connection, symmetric identities, summation formulae. 2020 MSC: 33E20, 33C45, 33B10, 33E30, 11T23. ©2024 All rights reserved. 1. Introduction and preliminaries Special polynomials refer to a class of polynomials that exhibit unique properties or have specific significance in various mathematical contexts. Examples of special polynomials include well-known fam- ilies such as Legendre polynomials, Chebyshev polynomials, Hermite polynomials, Bell polynomials, Touchard polynomials, Hermite polynomials and others. These polynomials often arise in mathematical physics, engineering, computer science, and other scientific disciplines, see for instance [1, 3, 4, 6, 8, 18– 20, 22]. The research on general cases of special polynomials has led to the discovery of new properties as- sociated with these polynomials. This implies that mathematicians have identified novel characteristics, relationships, or applications of special polynomials that were not previously known. These discoveries * Corresponding author Email addresses: shahidwani177@gmail.com (Shahid Ahmad Wani), w.ramirezquiroga@students.uninettunouniversity.net (William Ram´ırez) doi: 10.22436/jmcs.035.03.03 Received: 2024-02-21 Revised: 2024-04-01 Accepted: 2024-04-15