Mangontarum Mahid M, Macodi-Ringia Amila P, Abdulcarim Normalah S
Department of Mathematics, Mindanao State University, Main Campus, 9700 Marawi City, Philippines.
Int Sch Res Notices. 2014 Dec 22;2014:678408. doi: 10.1155/2014/678408. eCollection 2014.
More properties for the translated Whitney numbers of the second kind such as horizontal generating function, explicit formula, and exponential generating function are proposed. Using the translated Whitney numbers of the second kind, we will define the translated Dowling polynomials and numbers. Basic properties such as exponential generating functions and explicit formula for the translated Dowling polynomials and numbers are obtained. Convexity, integral representation, and other interesting identities are also investigated and presented. We show that the properties obtained are generalizations of some of the known results involving the classical Bell polynomials and numbers. Lastly, we established the Hankel transform of the translated Dowling numbers.
提出了第二类平移惠特尼数的更多性质，如水平生成函数、显式公式和指数生成函数。利用第二类平移惠特尼数，我们将定义平移道林多项式和数。得到了平移道林多项式和数的指数生成函数和显式公式等基本性质。还研究并给出了凸性、积分表示和其他有趣的恒等式。我们表明所得到的性质是涉及经典贝尔多项式和数的一些已知结果的推广。最后，我们建立了平移道林数的汉克尔变换。
