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.
Citation: William Ramírez, Can Kızılateş, Clemente Cesarano, Maria-Fernanda Heredia-Moyano, Talha Usman. Some properties of mixed-type hypergeometric Bernoulli/Euler $ r $-Dowling polynomials and their zero distribution patterns[J]. Networks and Heterogeneous Media, 2026, 21(6): 2223-2245. doi: 10.3934/nhm.2026083
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.
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Figures(3) / Tables(3)
William Ramírez, Can Kızılateş, Clemente Cesarano, Maria-Fernanda Heredia-Moyano, Talha Usman. Some properties of mixed-type hypergeometric Bernoulli/Euler $ r $-Dowling polynomials and their zero distribution patterns[J]. Networks and Heterogeneous Media, 2026, 21(6): 2223-2245. doi: 10.3934/nhm.2026083
Distributions of the zeros of the mixed-type hypergeometric Bernoulli $ r $-Dowling polynomials $ {}_B \mathcal{D}_{n}^{[m-1]}(x; \mu, r) $ and the mixed-type hypergeometric Euler $ r $-Dowling polynomials $ {}_E \mathcal{D}_{n}^{[m-1]}(x; \mu, r) $ corresponding to $ n = 10 $, $ n = 25, $ and $ n = 40 $. (a) $ {}_B \mathcal{D}_{n}^{[m-1]}(x; \mu, r) $ for $ n = 10 $, (b) $ {}_E \mathcal{D}_{n}^{[m-1]}(x; \mu, r) $ for $ n = 10 $, (c) $ {}_B \mathcal{D}_{n}^{[m-1]}(x; \mu, r) $ for $ n = 25 $, (d) $ {}_E \mathcal{D}_{n}^{[m-1]}(x; \mu, r) $ for $ n = 25 $, (e) $ {}_B \mathcal{D}_{n}^{[m-1]}(x; \mu, r) $ for $ n = 40 $, and (f) $ {}_E \mathcal{D}_{n}^{[m-1]}(x; \mu, r) $ for $ n = 40 $
Mesh distributions of the zeros of the mixed-type hypergeometric Bernoulli $ r $-Dowling polynomials $ {}_B \mathcal{D}_{n}^{[m-1]}(x; \mu, r) $ and the mixed-type hypergeometric Euler $ r $-Dowling polynomials $ {}_E \mathcal{D}_{n}^{[m-1]}(x; \mu, r) $ corresponding to $ n = 10 $, $ n = 25 $, and $ n = 40 $. (a) $ {}_B \mathcal{D}_{n}^{[m-1]}(x; \mu, r) $ for $ n = 10 $, (b) $ {}_E \mathcal{D}_{n}^{[m-1]}(x; \mu, r) $ for $ n = 10 $, (c) $ {}_B \mathcal{D}_{n}^{[m-1]}(x; \mu, r) $ for $ n = 25 $, (d) $ {}_E \mathcal{D}_{n}^{[m-1]}(x; \mu, r) $ for $ n = 25 $, (e) $ {}_B \mathcal{D}_{n}^{[m-1]}(x; \mu, r) $ for $ n = 40 $, and (f) $ {}_E \mathcal{D}_{n}^{[m-1]}(x; \mu, r) $ for $ n = 40 $
Three-dimensional mesh distribution of (a) the mixed-type hypergeometric Bernoulli $ r $-Dowling polynomials $ {}_B \mathcal{D}_{n}^{[m-1]}(x; \mu, r) $ and (b) the mixed-type hypergeometric Euler $ r $-Dowling polynomials $ {}_E \mathcal{D}_{n}^{[m-1]}(x; \mu, r) $, corresponding to $ n = 40 $
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