AI 中文总结
研究摩根 - 沃伊斯型多项式，通过引入新多项式族建立结构性质，定义变体证明相关和的表达式及推导生成函数，构建欧拉 - 塞德尔矩阵，得到新代数恒等式，扩展矩阵方法在组合分析中的应用。
AI 中文摘要
本文搭建了退化特殊多项式、欧拉 - 塞德尔矩阵方法和摩根 - 沃伊斯多项式领域之间的桥梁。引入了两个新的摩根 - 沃伊斯型多项式族并建立其结构性质，包括显式公式和递推关系。定义了三个多项式变体，证明了贝尔多项式的三个不同二项式型和可表示为涉及这些新族的有限和并推导其指数生成函数。还构建了欧拉 - 塞德尔矩阵，得到新代数恒等式并扩展了矩阵方法在组合分析中的应用。
英文摘要
This paper bridges the domains of degenerate special polynomials, the Euler-Seidel matrix method, and Morgan-Voyce polynomials. We introduce two new families of Morgan-Voyce type polynomials and establish their structural properties, including explicit formulas and recurrence relations. Additionally, we define three polynomial variants and prove that three distinct binomial-type sums for Bell polynomials can be expressed as finite sums involving these new families, and derive their exponential generating functions. We then construct Euler-Seidel matrices using the initial sequences associated with Morgan-type polynomials. Our results yield novel algebraic identities and expand the application of matrix methods in combinatorial analysis.
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