Algebraic properties of central Bell-based type 2 Bernoulli and Euler polynomials of complex variable

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Abstract

Recently, by combining type 2 Bernoulli and Euler polynomials with the central Bell polynomials, the central Bell-based type 2 Bernoulli and Euler polynomials of order α were considered, and many of their properties, formulas, and applications were investigated. The main aim of this work is to consider higher-order central Bell-based type 2 Bernoulli and Euler polynomials of complex variable, by which, both sine and cosine central Bell-based type 2 Bernoulli and Euler polynomials of order λ are introduced by treating the imaginary and real components separately. Then, diverse summation formulas, differential formulas, addition formulas, and correlation formulas with new and existing old polynomials and numbers are derived in a systematic way. Also, several intriguing connections of sine and cosine central Bell-based type 2 Bernoulli and Euler polynomials of order λ with the bivariate and one-variable central Bell polynomials, and the classical Stirling and central factorial numbers of the second kinds are investigated in detail. Moreover, the first few members of the new polynomials are provided by the lists, and the distributions of zeros of the new polynomials are illustrated by graphical representations, enhancing the understanding of the numerical data and facilitating a more intuitive grasp of the concepts discussed.

Keywords

type 2 Euler polynomials / type 2 Bernoulli polynomials / central Bell polynomials / partial differential equations / differential equations / process innovation / sine and cosine polynomials

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