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  • Open Access


    A Note on Bell-Based Bernoulli and Euler Polynomials of Complex Variable

    N. Alam1, W. A. Khan2,*, S. Obeidat1, G. Muhiuddin3, N. S. Diab1, H. N. Zaidi1, A. Altaleb1, L. Bachioua1

    CMES-Computer Modeling in Engineering & Sciences, Vol.135, No.1, pp. 187-209, 2023, DOI:10.32604/cmes.2022.021418

    Abstract In this article, we construct the generating functions for new families of special polynomials including two parametric kinds of Bell-based Bernoulli and Euler polynomials. Some fundamental properties of these functions are given. By using these generating functions and some identities, relations among trigonometric functions and two parametric kinds of Bell-based Bernoulli and Euler polynomials, Stirling numbers are presented. Computational formulae for these polynomials are obtained. Applying a partial derivative operator to these generating functions, some derivative formulae and finite combinatorial sums involving the aforementioned polynomials and numbers are also obtained. In addition, some remarks and observations on these polynomials are… More >

  • Open Access


    Some Properties of Degenerate r-Dowling Polynomials and Numbers of the Second Kind

    Hye Kyung Kim1,*, Dae Sik Lee2

    CMES-Computer Modeling in Engineering & Sciences, Vol.133, No.3, pp. 825-842, 2022, DOI:10.32604/cmes.2022.022103

    Abstract The generating functions of special numbers and polynomials have various applications in many fields as well as mathematics and physics. In recent years, some mathematicians have studied degenerate version of them and obtained many interesting results. With this in mind, in this paper, we introduce the degenerate r-Dowling polynomials and numbers associated with the degenerate r-Whitney numbers of the second kind. We derive many interesting properties and identities for them including generating functions, Dobinski-like formula, integral representations, recurrence relations, differential equation and various explicit expressions. In addition, we explore some expressions for them that can be derived from repeated applications… More >

  • Open Access


    Partial Bell Polynomials, Falling and Rising Factorials, Stirling Numbers, and Combinatorial Identities

    Siqintuya Jin1, Bai-Ni Guo2,*, Feng Qi3,*

    CMES-Computer Modeling in Engineering & Sciences, Vol.132, No.3, pp. 781-799, 2022, DOI:10.32604/cmes.2022.019941

    Abstract In the paper, the authors collect, discuss, and find out several connections, equivalences, closed-form formulas, and combinatorial identities concerning partial Bell polynomials, falling factorials, rising factorials, extended binomial coefficients, and the Stirling numbers of the first and second kinds. These results are new, interesting, important, useful, and applicable in combinatorial number theory. More >

  • Open Access


    Degenerate s-Extended Complete and Incomplete Lah-Bell Polynomials

    Hye Kyung Kim1,*, Dae Sik Lee2

    CMES-Computer Modeling in Engineering & Sciences, Vol.131, No.3, pp. 1479-1495, 2022, DOI:10.32604/cmes.2022.017616

    Abstract Degenerate versions of special polynomials and numbers applied to social problems, physics, and applied mathematics have been studied variously in recent years. Moreover, the (s-)Lah numbers have many other interesting applications in analysis and combinatorics. In this paper, we divide two parts. We first introduce new types of both degenerate incomplete and complete s-Bell polynomials respectively and investigate some properties of them respectively. Second, we introduce the degenerate versions of complete and incomplete Lah-Bell polynomials as multivariate forms for a new type of degenerate s-extended Lah-Bell polynomials and numbers respectively. We investigate relations between these polynomials and degenerate incomplete and… More >

  • Open Access


    Bell Polynomial Approach for the Solutions of Fredholm Integro-Differential Equations with Variable Coefficients

    Gökçe Yıldız1, Gültekin Tınaztepe2, *, Mehmet Sezer1

    CMES-Computer Modeling in Engineering & Sciences, Vol.123, No.3, pp. 973-993, 2020, DOI:10.32604/cmes.2020.09329

    Abstract In this article, we approximate the solution of high order linear Fredholm integro-differential equations with a variable coefficient under the initial-boundary conditions by Bell polynomials. Using collocation points and treating the solution as a linear combination of Bell polynomials, the problem is reduced to linear system of equations whose unknown variables are Bell coefficients. The solution to this algebraic system determines the approximate solution. Error estimation of approximate solution is done. Some examples are provided to illustrate the performance of the method. The numerical results are compared with the collocation method based on Legendre polynomials and the other two methods… More >

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