Promotion, Generalized Stirling Numbers

The Stirling numbers were first introduced by J. Stirling in his book Methodus Differentialis (1730). Since then, many authors paid lots of interests on it. A thorough presentation of Stirling numbers and their most important properties provided by Ch. Jordan (1933) in an excellent paper. The term "Generalized Striding Numbers" first appeared in L. Comtet's paper [4]. The authors like L. C. Hsu had deeply studied on it.By studying Generalized Stirling Numbers, we extend their domains in this paper.In Chapter 1, the notation and properties of Striding numbers and generalized Strirling numbers are introduced.In Chapter 2, we extend the recurrence relations of Generalized Strirling Numbers. And in term of this, we extend the domains of Generalized Strirling Gumbers to negative integers. We study the properties of Generalized Strirling Numbers,introduced "principle branch" and "Taylor branch", and provided some explicit formulas.In Chapter 3, we study the cube models on Generalized Strirling Numbers.These models provided combinatorial interpretations of Generalized Strirling Numbers. We extend these models and derive some analogical results presented in Chapter 1.In Chapter 4, we make a preliminary study on Generalized Strirling Numbers with multi parameters.