Some properties of mixed-type hypergeometric Bernoulli/Euler r-Dowling polynomials and their zero distribution patterns

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Abstract

In this paper, we introduce a new class of polynomials called the mixed-type hypergeometric Bernoulli and Euler r-Dowling polynomials. These polynomials extend the classical Bernoulli and Euler polynomials, together with their hypergeometric (level-m) versions, while incorporating the combinatorial structure of the r-Dowling framework within a single generating function. Using exponential generating functions, we derive their main properties such as explicit formulas, recurrence relations, differential identities, and integral representations. In addition, we examine the zeros of these polynomials numerically and graphically. The results show that the zeros follow symmetric and structured patterns in the complex plane and display regular behavior as the degree increases. Overall, this study provides a natural extension of classical polynomial families and offers new insights for further research in approximation theory and related areas.

Keywords

r-Dowling polynomials / hypergeometric Bernoulli and Euler polynomials / mixed-type polynomials

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