EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 3, Article Number 6054 ISSN 1307-5543 – ejpam.com Published by New York Business Global An Analytical Study of Two-Dimensional Bell Polynomials and Their Properties Shahid AhmadWani1, Taghreed Alqurashi2, William Ramı́rez3,4,∗, Shilpa Malge1, Jesús David Berŕıo Valbuena5 1Symbiosis Institute of Technology PUNE, Symbiosis International (Deemed University), Pune, India 2Mathematics Department, Faculty of Science, Al-Baha University, 65779-7738 Albaha city, Kingdom of Saudi, Arabia 3 Department of Natural and Exact Sciences, Universidad de la Costa, Calle 58 N 55-66, 080002 Barranquilla, Colombia 4 Section of Mathematics International Telematic University Uninettuno, Corso Vittorio Emanuele II, 39, 00186 Rome, Italy 5 Universidad del Atlántico, Barranquilla, Colombia Abstract. This study investigates two-dimensional Bell polynomials, emphasizing their funda- mental properties and applications in mathematical analysis. Utilising the framework of generat- ing functions, we derive explicit representations, summation formulae, recurrence relations, and addition formulas for these polynomials. Furthermore, we introduce the 2D Bell-based Stirling polynomials of the second kind and explore their associated properties. This research aims to enhance the theoretical understanding of Bell polynomials and their broader applications in math- ematical analysis. 2020 Mathematics Subject Classifications: 33E20, 33C45, 33B10, 33E30, 11T23 Key Words and Phrases: 2D Special Polynomials, Generating function, Explicit form, Series representation 1. Introduction and preliminaries A fascinating class of mathematical functions, namely special polynomials, are char- acterized by unique properties and find specific significance in various mathematical con- texts, for example [1–5]. These polynomials encompass well-known families such as Leg- endre polynomials, Chebyshev polynomials, Hermite polynomials, Bell polynomials, and Touchard polynomials. Legendre polynomials, for example, arise in problems involving ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6054 Email addresses: shahidwani177@gmail.com (S.A. Wani), talqorashi@bu.edu.sa (T. Alqurashi), W. Ramı́rez (wramirez4@cuc.edu.co), shilpam@sitpune.edu.in ( S. Malge), jberriovalbuena@mail.uniatlantico.edu.co (J. Berŕıo) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) S. A. Wani et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6054 2 of 16 electrostatics and fluid dynamics, while Chebyshev polynomials have applications in nu- merical analysis and signal processing. On the other hand, Hermite polynomials frequently emerge in quantum mechanics and probability theory. Bell and Touchard polynomials also play important roles in combinatorics and number theory. Studying these special polyno- mials and their applications is essential in mathematical physics, engineering, computer science, and other scientific disciplines. On the topic of polynomial families and their vari- ous extensions, much research has appeared in the literature (see, for example, [8–10, 13]). The exploration of special polynomials in general cases has revealed new properties and applications, greatly expanding our mathematical understanding of these polynomials. Mathematicians have discovered unique characteristics, relationships, and applications that were previously unknown, enriching the field of mathematics. The study of special polynomials is crucial due to their frequent appearance in solving “differential equations, orthogonal polynomial theory, numerical analysis, and various other mathematical and computational problems”. These polynomials exhibit specific algebraic structures and recurrence relations, which make them particularly amenable to analysis, enhancing the broader study of algebra and mathematical structures. Additionally, special polynomials have profound connections to other areas of mathematics, such as “combinatorics, number theory, and analysis”. These interconnections promote interdisciplinary research and the advancement of mathematical theories, leading to broader implications across scientific and applied fields. Consequently, the ongoing research and discoveries in the properties of special polynomials continue to play a vital role in advancing both pure and applied mathematics. One of the most intriguing and significant classes of polynomial sequences and numbers is the Stirling numbers. These numbers form a family that is essential in combinatorics, particularly in problems related to permutations, combinations, and partitions, for exam- ple, [6, 7, 14–19]. There are two primary types of Stirling numbers, each serving a distinct purpose in combinatorial mathematics: the Stirling numbers of the first kind, denoted as S1(n, ϵ), and the Stirling numbers of the second kind, denoted as S2(n, ϵ). The Stirling numbers of the first kind, S1(n, ϵ), represent the number of permutations of n elements that contain exactly ϵ cycles. This means they count the number of ways to arrange n distinct elements into ϵ cyclic groups. On the other hand, the Stirling numbers of the second kind, S2(n, ϵ), quantify the number of ways to partition a set of n distinct elements into ϵ non-empty, indistinguishable subsets, where the order of these subsets does not matter. These numbers are significant in various mathematical and applied fields, includ- ing combinatorics, where they facilitate the understanding and solving problems related to permutations and partitions. Their utility extends to areas such as algebra, probabil- ity, and the analysis of algorithms, underscoring their importance in both theoretical and practical applications of mathematics. Stirling numbers are widely used in combinatorics for counting permutations, combinations, and partitions. “Stirling Polynomials of the Second Kind”, denoted as S2(n, ϵ; q1), are associated with exponential generating functions. The exponential generating function for Stirling poly- S. A. Wani et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6054 3 of 16 nomials of the second kind is given by: ∞∑ n=0 S2(n, ϵ; q1) ξ n n! = (eξq1 − 1)ϵ ϵ! . (1) Certainly, the exponential generating function for Stirling numbers of the second kind simplifies to the following expression when q1 = 1: ∞∑ n=0 S2(n, ϵ) ξ n n! = (eξ − 1)ϵ ϵ! . (2) Further, the recurrence relation for Stirling numbers of the second Kind S2(n, ϵ) can be computed using the recurrence relation: qn1 = ∞∑ n=0 S2(n, ϵ) (q1)ϵ (3) or (q1)n = n∑ ϵ=0 S2(n, ϵ) q ϵ 1, (4) where, the falling factorial is given by (q1)ϵ = q1(q1 − 1)(q2 − 2) · · · (q1 − (ϵ− 1)). Additionally, for every non-negative integer ϵ in the set of natural numbers, the fol- lowing expression holds true: Sϵ(n) = n∑ l=0 lϵ. The sum of integer powers is referred to as the “sum of integer powers”, and the exponential generating function for Sϵ(n) is as follows: ∞∑ ϵ=0 Sϵ(n) ξϵ ϵ! = e(n+1)ξ−1 eξ − 1 . (5) The concepts mentioned are crucial in combinatorics and are widely used to solve counting problems involving the organisation of distinguishable objects into partitions and subsets. The incredible power of exponential operators shines through, especially when solving differential equations. These operators simplify the analysis and offer a convenient way to express solutions. Bell’s groundbreaking work (Bell, 2010) presents a comprehensive exploration of the foundational formalism. It brilliantly demonstrates that by making a suitable change of variable, the effect of the operator on a given function of q1 can be viewed as that of a traditional shift operator. In other words, for any parameter µ, applying the shift operator exp(µ∂q1) to any function of q1 yields the following remarkable result: exp(µ∂q1){f(q1)} = ∞∑ n=0 ∂n q1f(q1) µn n! = ∞∑ n=0 fn(q1) µn n! = f(q1 + µ), (6) S. A. Wani et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6054 4 of 16 where ∂n q1 = ∂n ∂qn1 . This result illustrates that the operator exp(µ∂q1) effectively shifts the argument of the function f by µ, simplifying the manipulation and solution of differential equations by transforming them into algebraic problems. The following identities are exploited from (6): exp(µ q21∂q1){f(q1)} = f ( q1 1− µq1 ) , (7) exp(µ∂q1){qn1 } = ( q1 + µ )n , exp(µ∂n q1){e q1} = eq1+µ, exp(µq1 ∂q1)f{q1} = f(eq1µ). One important class of special polynomials is the Bell polynomials [1], named after mathematician Eric Temple Bell. Bell polynomials play a crucial role in representing the partial Bell polynomials, which correspond to the partial derivatives of the exponen- tial generating function. These polynomials have wide-ranging applications in fields such as combinatorics, probability theory, and the analysis of algorithms. Bell polynomials are particularly valuable for counting and enumerating various combinatorial structures in combinatorics. They describe partitions of sets, which involve dividing a set into non- overlapping subsets, and compositions of integers, where an integer is expressed as the sum of ordered integers. The utility of Bell polynomials extends to the analysis of algorithms, where they help in understanding the performance and behaviour of combinatorial algo- rithms. Notable works that explore these applications include references [11, 12, 14–19], which delve into the diverse and significant uses of Bell polynomials in these mathemat- ical domains. Through these applications, Bell polynomials demonstrate their versatility and importance in solving complex problems and providing insights across various areas of mathematical research. These polynomials are a sequence that arises in combinatorics and is denoted as B [j] n (q1). The following exponential generating function defines them: ∞∑ n=0 B[j] n (q1) ξn n! = eq1(e ξ−1). (8) For the case where q1 = 1, the Bell polynomials simplify to the Bell numbers according to the following relation: ∞∑ n=0 B[j] n ξn n! = ee ξ−1. Bell polynomials are an incredibly versatile and powerful framework that offers deep insights into combinatorial structures, particularly in partitioning, generating functions, and algebraic combinatorics. Their significance in mathematics cannot be overstated, as they provide an effective tool for counting problems and analyzing discrete structures. S. A. Wani et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6054 5 of 16 Here, we have made a significant advancement in the field by developing a new formulation for the generating relation of 2D Bell polynomials given by the generating expression: ∞∑ n=0 B[j] n (q1, q2) ξn n! = eq1(e ξ−1)+q2(eξ−1)j . (9) For, q2 = 0 in (9), the 2D Bell polynomials B [j] n (q1, q2) reduce to the Bell polynomials given by (8). In the upcoming sections, we will explore these mathematical marvels, un- covering their intricate properties and unveiling their remarkable applications. In Section 2, we dive headfirst into the realm of generating functions, where we introduce 2D Bell polynomials. Further, we derive explicit representations and unveil summation formulae, recurrence relations, and addition formulas, all while shedding light on their profound connection to Stirling polynomials of the second kind. In Section 3, we delve deeper into these polynomials’ matrix form and product formula, unveiling their structural insights into their inner workings. In Section 4, where we introduce the 2D Bell-based Stirling polynomials of the second kind, expanding the horizons of our understanding even fur- ther. The conclusion is provided last. For j = 3, the first five 2D Bell polynomials are as follows: B [3] 0 (q1, q2) = 1, B [3] 1 (q1, q2) = q1, B [3] 2 (q1, q2) = q21 + q1, B [3] 3 (q1, q2) = q31 + 3q21 + q1 + 6q2, B [j] 4 (q1, q2) = q41 + 6q31 + 7q21 + q1 + 36q2 + 24q1q2. Figure 1: B[3] 3 (q1, q2) = q31 + 3q21 + q1 + 6q2 Figure 2: B[4] 3 (q1, q2) = q41 + 6q31 + 7q21 + q1 + 36q2 + 24q1q2 S. A. Wani et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6054 6 of 16 2. 2D Bell polynomials The 2D special Bell polynomials play a crucial role in combinatorial analysis, number theory, and statistical mechanics. They express multivariate exponential generating func- tions and find applications in studying combinatorial structures and problems in discrete mathematics. Additionally, they are used in number theory to investigate properties of partitions, compositions, and other combinatorial objects. In statistical mechanics, these polynomials are employed to analyze the behavior of systems with two-dimensional degrees of freedom, providing insights into the thermodynamic properties and phase transitions of physical systems. Furthermore, in applied mathematics and engineering, 2D special Bell polynomials are utilized in problems involving complex systems with multiple variables, offering a systematic framework for modelling and analysis. Overall, the significance of 2D special Bell polynomials lies in their ability to bridge theoretical concepts with practi- cal applications across diverse disciplines, making them invaluable tools for mathematical research and problem-solving in various fields. Here, in this section, we derive the explicit forms and certain other properties of 2D Bell polynomials denoted by B [j] n (q1, q2) as follows: Theorem 1. The 2D Bell polynomials denoted by B [j] n (q1, q2) satisfy the listed explicit form: B[j] n (q1, q2) = [n]∑ s=0 ( n s ) B [j] n−s(q1) S2(s, k) (e ξ − 1)jk−k 1− q2 . (10) Proof. The expression denoted by (9) can be expressed in view of the identity eA+B = eAeB, as eq1(e ξ−1)+q2(eξ−1)j = eq1(e ξ−1)eq2(e ξ−1)j = ( ∞∑ n=0 B[j] n (q1) ξn n! )( ∞∑ r=0 qr2 r! (eξ − 1)jr ) = ∞∑ n=0 B[j] n (q1, q2) ξn n! . Expand the second exponential using Stirling numbers given by expression (2), and (eξ − 1)jr can be written using a convolution involving Stirling numbers: (eξ−1)jr = ∑∞ s=0 S2(s, k) (e ξ− 1)jk−k ξs s! , , it follows that eq1(e ξ−1)+q2(eξ−1)j = ∞∑ n=0 B[j] n (q1, q2) ξn n! = ( ∞∑ n=0 B[j] n (q1) ξn n! )( ∞∑ s=0 S2(s, k) (e ξ − 1)jk−k 1− q2 ξs s! ) . Applying the Cauchy product for power series:( ∞∑ n=0 an ξn n! )( ∞∑ s=0 bs ξs s! ) = ∞∑ n=0 ( n∑ s=0 ( n s ) an−sbs ) ξn n! . in preceding expression, it follows that ∞∑ n=0 B[j] n (q1, q2) ξn n! = ∞∑ n=0 ( n∑ s=0 ( n s ) B [j] n−s(q1) S2(s, k) (e ξ − 1)jk−k 1− q2 ) ξn n! . S. A. Wani et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6054 7 of 16 By comparing the coefficients of ξn n! on both sides of the original generating function, we conclude: B[j] n (q1, q2) = n∑ s=0 ( n s ) B [j] n−s(q1) S2(s, k) (e ξ − 1)jk−k 1− q2 . This completes the proof. Theorem 2. The Bell polynomials B [j] n (q1, q2), have a series representation as listed be- low: B[j] n (q1, q2) = [n]∑ s=0 ( n s ) S2(n− s, l) 1− q1 S2(s, k) (e ξ − 1)jk−k 1− q2 . (11) Proof. The expression denoted by (9) can be expressed in the form eq1(e ξ−1)+q2(eξ−1)j = eq1(e ξ−1)eq2(e ξ−1)j . Using the expression (1)–(4) into the right-hand side of the previous expression, we deter- mine eq1(e ξ−1)+q2(eξ−1)j = ∞∑ l=0 ∞∑ n=0 q1 lS2(n, l) ξn n! (eξ − 1)jk−kn! ∞∑ r=0 ∞∑ s=0 q2 rS2(s, k) ξs s! (eξ − 1)jk−k. Placing the right-hand side of the equation (9) into the left-hand side of the preceding expression and then simplifying the right-hand side, we can conclude that ∞∑ n=0 B[j] n (q1, q2) ξn n! = ∞∑ n=0 ∞∑ s=0 S2(n, l) 1− q1 S2(s, k) (e ξ − 1)jk−k 1− q2 ξn+s n! s! . Rearranging the series, we can substitute n − s for n into the right-hand side of the preceding expression and then simplifying the right-hand side, we can conclude that ∞∑ n=0 B[j] n (q1, q2) ξn n! = ∞∑ n=0 [n∑ s=0 ( n s ) S2(n− s, l) 1− q1 S2(s, k)(e ξ − 1)jk−k 1− q2 ξn n! . While comparing the similar abilities of ξn n! in the preceding statement, we arrive at state- ment (11). Theorem 3. The 2D Bell polynomials denoted by B [j] n (q1, q2). Then the following sum- mation formulas hold. B[j] n (q1 + q3, q2) = n∑ k=0 ( n k ) B [j] n−k(q1, q3)B [j] k (q2). S. A. Wani et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6054 8 of 16 Proof. By (9) and (8), we have ∞∑ n=0 B[j] n (q1 + q3, q2) ξn n! = e(q1+q2)(eξ−1)+q3(eξ−1)j = ∞∑ n=0 B[j] n (q1, q3) ξn n! ∞∑ k=0 B [j] k (q2) ξk k! = ∞∑ n=0 [ n∑ k=0 ( n k ) B [j] n−k(q1, q3)B [j] k (q2) ] ξn n! . Finally equating the coefficients of ξn n! of both sides, we get the asserted Theorem 8. Theorem 4. For any arbitrary n ∈ N, the following relation hold true: B[j] n (q1 + 1, q2)− B[j] n (q1, q2) = n∑ k=0 ( n k ) B [j] n−k(q1, q2)B [j] k − B[j] n (q1, q2). (12) Proof. Utilizing the expression (9), we find ∞∑ n=0 [ B[j] n (q1 + 1, q2)− B[j] n (q1, q2) ] ξn n! = e(q1+1)(eξ−1)+q2(eξ−1)j − eq1(e ξ−1)+q2(eξ−1)j = eq1(e ξ−1)+q2(eξ−1)j [ ee ξ−1 − 1 ] = ∞∑ n=0 [ n∑ k=0 ( n k ) B [j] n−k(q1, q2)Bn(q1, q2)Bk ] ξn n! . By equating both sides, we obtained the result (12). Theorem 5. For n ≥ 1, let B [j] n (q1, q2) be the 2D Bell polynomials. Then we have ∂ ∂q1 B[j] n (q1, q2) = 1 (n2 + n) n∑ k=0 ( n+ 1 k ) B [j] k (q1, q2) (13) and ∂ ∂q2 B[j] n (q1, q2) = n∑ k=0 ( n k ) B [j] n−k(q1, q2)j!S2(k, j). (14) Proof. (See (13)). Differentiating partially with respect to the variable q1 on both sides of the generating function: ∞∑ n=0 B[j] n (q1, q2) ξn n! = eq1(e ξ−1)+q2(eξ−1)j , S. A. Wani et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6054 9 of 16 we get ∂ ∂q1 [ ∞∑ n=0 B[j] n (q1, q2) ξn n! ] = ∂ ∂q1 [ eq1(e ξ−1)+q2(eξ−1)j ] = eq1(e ξ−1)+q2(eξ−1)j (eξ − 1). Using the generating function of B [j] n (q1, q2), we substitute: eq1(e ξ−1)+q2(eξ−1)j (eξ − 1) = [ ∞∑ n=0 B[j] n (q1, q2) ξn n! ][ ∞∑ r=1 ξr r! ] = ∞∑ n=0 n∑ k=0 ( n+ 1 k ) B [j] k (q1, q2) ξn+1 (n+ 1)! . Equating the coefficients of ξn n! on both sides gives: ∂ ∂q1 B[j] n (q1, q2) = 1 (n2 + n) n∑ k=0 ( n+ 1 k ) B [j] k (q1, q2). Proof. (See (14)). Differentiating partially with respect to the variable q2 on both sides of the generating function: ∞∑ n=0 B[j] n (q1, q2) ξn n! = eq1(e ξ−1)+q2(eξ−1)j , we get ∂ ∂q2 [ ∞∑ n=0 B[j] n (q1, q2) ξn n! ] = ∂ ∂q2 [ eq1(e ξ−1)+q2(eξ−1)j ] = eq1(e ξ−1)+q2(eξ−1)j (eξ − 1)j . Recall that the exponential generating function of j!S2(n, j) is: (eξ − 1)j = ∞∑ n=0 j!S2(n, j) ξn n! . So, we compute: eq1(e ξ−1)+q2(eξ−1)j (eξ − 1)j = [ ∞∑ n=0 B[j] n (q1, q2) ξn n! ][ ∞∑ m=0 j!S2(m, j) ξm m! ] = ∞∑ n=0 n∑ k=0 ( n k ) B [j] n−k(q1, q2)j!S2(k, j) ξn n! . S. A. Wani et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6054 10 of 16 Equating coefficients of ξn n! yields: ∂ ∂q2 B[j] n (q1, q2) = n∑ k=0 ( n k ) B [j] n−k(q1, q2)j!S2(k, j). Theorem 6. For n ≥ 0, let { B [j] n (q1, q2) } n≥0 be the sequences of 2D Bell polynomials in the variable q1, q2 and q3, they satisfy the following relation n∑ k=0 ( n k )[ B [j] k (q1 + q3, q2)B [j] n−k(2q2)− B [j] n−k(q1, q2)B [j] n (q3, q2) ] = 0. Proof. Let’s consider the following expressions eq1(e ξ−1)+q2(eξ−1)j = ∞∑ n=0 B[j] n (q1, q2) ξn n! (15) and eq3(e ξ−1)+q2(eξ−1)j = ∞∑ n=0 B[j] n (q3, q2) ξn n! . (16) From (15) and (16), we have e(q1+q3)(eξ−1)+q2(eξ−1)je2q2(e ξ−1)j = ( ∞∑ n=0 B[j] n (q1, q2) ξn n! )( ∞∑ n=0 B[j] n (q3, q2) ξn n! ) ( ∞∑ n=0 B[j] n (q1 + q3, q2) ξn n! )( ∞∑ n=0 B[j] n (0, 2q2) ξn n! ) = ( ∞∑ n=0 B[j] n (q1, q2) ξn n! )( ∞∑ n=0 B[j] n (q3, q2) ξn n! ) ∞∑ n=0 n∑ k=0 ( n k ) B [j] k (q1 + q3, q2)B [j] n−k(2q2) ξn n! = ∞∑ n=0 n∑ k=0 ( n k ) B [j] n−k(q1, q2)B [j] n (q3, q2) ξn n! n∑ k=0 ( n k ) B [j] k (q1 + q3, q2)B [j] n−k(2q2) = n∑ k=0 ( n k ) B [j] n−k(q1, q2)B [j] n (q3, q2). Therefore, n∑ k=0 ( n k )[ B [j] k (q1 + q3, q2)B [j] n−k(2q2)− B [j] n−k(q1, q2)B [j] n (q3, q2) ] = 0. S. A. Wani et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6054 11 of 16 3. The 2D Bell-based Stirling polynomials of the second kind Definition 1. ∞∑ n=0 BS [j] 2 (n, ϵ; q1, q2) ξn n! = (eξ − 1)ϵ ϵ! eq1(e ξ−1)+q2(eξ−1)j . (17) This description serves as a foundational concept, establishing the basis for additional exploration and comprehension of these polynomials’ important implications and possible applications in the broader realm of mathematics. For ϵ, j = 2, the first four two-variable Bell-based Stirling polynomials are as follows: BS [2] 2 (0, 2; q1, q2) = 1 2 , BS [2] 2 (1, 2; q1, q2) = 1 2 q1 + 1 2 , BS [2] 2 (2, 2; q1, q2) = 1 2 q21 + 3 2 q1 + q2 + 7 12 , BS [2] 2 (3, 2; q1, q2) = 1 2 q31 + 3q21 + 3q1q2 + 6q2 + 3 4 . Figure 3: 1 2 q21 + 3 2 q1 + q2 + 7 12 Figure 4: 1 2 q31 + 3q21 + 3q1q2 + 6q2 + 3 4 Remark 1. The expression given by (17) yields a set of polynomials called the Bell- Stirling polynomials of the second kind when we substitute q2 = 0. This set of polynomials is expressed as: ∞∑ n=0 BS2(n, ϵ; q1) ξn n! = (eξ − 1)ϵ ϵ! eq1(e ξ−1). Remark 2. After substituting q1 = q2 = 0 into the expression from (17), a group of polynomials called the Stirling numbers of the second kind, as shown in (2), is derived. S. A. Wani et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6054 12 of 16 Theorem 7. For any non-negative integer n, the Stirling polynomials of the second kind based on 2D Bell numbers exhibit the following correlation: n∑ l=0 ( n l ) S2(l, ϵ)Bn−l(q1, q2) = BS2(n, ϵ; q1, q2). Proof. The expression labelled as (17) can be expressed as: ∞∑ n=0 BS2(n, ϵ; q1, q2) ξn n! = (eξ − 1)ϵ ϵ! eq1(e ξ−1)+q2(eξ−1)j = ∞∑ n=ϵ S2(n, ϵ) ξn n! ∞∑ n=0 B[j] n (q1, q2) ξn n! , the above expression can be expressed in another form as ∞∑ n=0 BS2(n, ϵ; q1, q2) ξn n! = ∞∑ n=0 n∑ l=0 ( n l ) S2(l, ϵ) B [j] n−l(q1, q2) ξn n! . (18) We achieve the expected outcome by contrasting the exponents of identical powers of ξ. Remark 3. The correlation satisfied by the Bell-based Stirling polynomials of the second kind is obtained by substituting q2 = 0 into the expression given by (17) as: n∑ l=0 ( n l ) S2(l, ϵ)Bn−l(q1) = BS2(n, ϵ; q1), for a non-negative integer n. Theorem 8. The 2D Bell-based Stirling polynomials of the second kind can be obtained for a non-negative integer n. There are applicable summation formulas for these polynomials: BS [j] 2 (n, ϵ; q1 + q3, q2 + q4) = n∑ k=0 ( n k ) BS [j] 2 (n− k, ϵ; q1, q2)B [j] k (q3, q4). Proof. Let’s examine the generating functions provided in equations (17) and (9). As a result, we obtain ∞∑ n=0 BS [j] 2 (n, ϵ; q1 + q3, q2 + q4) ξn n! = (eξ − 1)ϵ ϵ! e(q1+q3)(eξ−1)+(q2+q4)(eξ−1)j = (eξ − 1)ϵ ϵ! eq1(e ξ−1)+q2(eξ−1)jeq3(e ξ−1)+q4(eξ−1)j S. A. Wani et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6054 13 of 16 = ∞∑ n=0 BS [j] 2 (n, ϵ; q1, q2) ξn n! ∞∑ n=0 B[j] n (q3, q4) ξn n! = ∞∑ n=0 n∑ k=0 ( n k ) BS [j] 2 (n− k, ϵ; q1, q2)B [j] k (q3, q4) ξn n! . At last, by setting the coefficients of ξn n! equal on both sides, we prove Theorem 8 as claimed. Theorem 9. The 2D Bell-based Stirling polynomials of the second kind should be consid- ered for a non-negative integer n. The following relation is valid: BS [j+β] 2 (n, ϵ; q1, q2) = n∑ k=0 ( n k ) BS [j] 2 (n− k, ϵ; q1, q2)B [β] k . Proof. By (17), we have ∞∑ n=0 BS [j+β] 2 (n, ϵ; q1, q2) ξn n! = (eξ − 1)ϵ ϵ! eq1(e ξ−1)+q2(eξ−1)j+β = (eξ − 1)ϵ ϵ! eq1(e ξ−1)+q2(eξ−1)je(e ξ−1)β = ∞∑ n=0 BS [j] 2 (n, ϵ; q1, q2) ξn n! ∞∑ n=0 B[β] n ξn n! = ∞∑ n=0 n∑ k=0 ( n k ) BS [j] 2 (n− k, ϵ; q1, q2)B [β] k ξn n! . Finally, by setting the coefficients of fracξnn! equal on both sides, we prove Theorem 9 as claimed. Theorem 10. For every n ≥ 1, if we let S [j] 2 (n, ϵ; q1, q2) represent the 2D Stirling polyno- mials of the second kind based on the Bell numbers, then the following holds: ∂ ∂q1 S [j] 2 (n, ϵ; q1, q2) = n∑ k=0 ( n k ) S [j] 2 (n− k, ϵ; q1, q2)− S [j] 2 (n, ϵ; q1, q2) (19) and ∂ ∂q2 S [j] 2 (n, ϵ; q1, q2) = n∑ k=0 ( n k ) S [j] 2 (n− k, ϵ; q1, q2)j!S2(k, j). (20) Proof. (see (19)). When we partially differentiate both sides of the equation (17) with respect to the variable q1, we get ∂ ∂q1 [ ∞∑ n=0 S [j] 2 (n, ϵ; q1, q2) ξn n! ] = ∂ ∂q1 [ (eξ − 1)ϵ ϵ! eq1(e ξ−1)+q2(eξ−1)j ] S. A. Wani et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6054 14 of 16 = (eξ − 1)ϵ ϵ! eq1(e ξ−1)+q2(eξ−1)j (eξ − 1) = ∞∑ n=0 S [j] 2 (n, ϵ; q1, q2) ξn n! ∞∑ n=0 ξn n! − ∞∑ n=0 S [j] 2 (n, ϵ; q1, q2) ξn n! = ∞∑ n=0 [ n∑ k=0 ( n k ) ∞∑ n=0 S [j] 2 (n− k, ϵ; q1, q2)− ∞∑ n=0 S [j] 2 (n, ϵ; q1, q2) ] ξn n! . So, ∂ ∂q1 S [j] 2 (n, ϵ; q1, q2) = n∑ k=0 ( n k ) S [j] 2 (n− k, ϵ; q1, q2)− S [j] 2 (n, ϵ; q1, q2). Proof. (See (20)). When we take the partial derivative with respect to the variable q2 of both sides of the equation (17), we get ∂ ∂q2 [ ∞∑ n=0 S [j] 2 (n, ϵ; q1, q2) ξn n! ] = ∂ ∂q1 [ (eξ − 1)ϵ ϵ! eq1(e ξ−1)+q2(eξ−1)j ] = (eξ − 1)ϵ ϵ! eq1(e ξ−1)+q2(eξ−1)j (eξ − 1)j = ∞∑ n=0 S [j] 2 (n, ϵ; q1, q2) ξn n! ∞∑ n=0 j!S2(n, j) ξn n! = ∞∑ n=0 n∑ k=0 ( n k ) S [j] 2 (n, ϵ; q1, q2)j!S2(k, j) ξn n! . So, ∂ ∂q2 S [j] 2 (n, ϵ; q1, q2) = n∑ k=0 ( n k ) S [j] 2 (n− k, ϵ; q1, q2)j!S2(k, j). 4. Conclusion In this article, we present 2D Bell polynomials using generating functions and thor- oughly examine their various associated properties. Our exploration includes explicit rep- resentations, summation formulae, recurrence relations, and addition formulas, providing valuable insights into their mathematical foundations. Lastly, we have introduced the 2D Bell-based Stirling polynomials of the second kind, broadening the scope of our study to include related concepts and results. Through this comprehensive analysis, our research contributes to a deeper comprehension of the properties and applications of Bell polyno- mials in mathematical analysis. This lays a solid groundwork for further exploration and practical use in diverse fields. Future research in the realm of 2D Bell polynomials could focus on several avenues to further expand our understanding and applications of these mathematical entities. One S. A. Wani et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6054 15 of 16 potential direction is the exploration of higher-dimensional generalizations beyond the 2D case, investigating how Bell polynomials can be extended to three or more variables and uncovering their properties and relationships in multi-dimensional spaces. Additionally, there is room for research into the development of more efficient compu- tational algorithms and numerical techniques for handling 2D Bell polynomials, especially in scenarios involving large datasets or complex systems. 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