N. Alam¹, W. A. Khan^2,*, 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 >

Lee-Chae Jang¹, Dae San Kim², Hanyoung Kim³, Taekyun Kim^3,*, Hyunseok Lee³

CMES-Computer Modeling in Engineering & Sciences, Vol.129, No.1, pp. 393-408, 2021, DOI:10.32604/cmes.2021.016917

Abstract Recently, degenerate poly-Bernoulli polynomials are defined in terms of degenerate polyexponential functions by Kim-Kim-Kwon-Lee. The aim of this paper is to further examine some properties of the degenerate poly-Bernoulli polynomials by using three formulas from the recently developed ‘λ-umbral calculus.’ In more detail, we represent the degenerate poly-Bernoulli polynomials by Carlitz Bernoulli polynomials and degenerate Stirling numbers of the first kind, by fully degenerate Bell polynomials and degenerate Stirling numbers of the first kind, and by higherorder degenerate Bernoulli polynomials and degenerate Stirling numbers of the second kind. More >

Taekyun Kim^1,*, Dae San Kim², Dmitry V. Dolgy³, Si-Hyeon Lee¹, Jongkyum Kwon^4,*

CMES-Computer Modeling in Engineering & Sciences, Vol.128, No.3, pp. 1121-1132, 2021, DOI:10.32604/cmes.2021.016532

Abstract We introduce the higher-order type 2 Bernoulli numbers and polynomials of the second kind. In this paper, we investigate some identities and properties for them in connection with central factorial numbers of the second kind and the higher-order type 2 Bernoulli polynomials. We give some relations between the higher-order type 2 Bernoulli numbers of the second kind and their conjugates. More >