Generalized Stirling Numbers and Generalized Stirling Functions

Details Generalized Stirling Numbers and Generalized Stirling Functions Generalized Stirling Numbers and Generalized Stirling Functions

2011-06-26

arxiv.org/abs/1106.5251v1

Mathematics

Combinatorics

26 pages, Presented on May 30, 2011, at City University of Hong Kong, International Conference on Asymptotics and Special Func

Scientific paper

Here presented is a unified approach to Stirling numbers and their generalizations as well as generalized Stirling functions by using generalized factorial functions, $k$-Gamma functions, and generalized divided difference. Previous well-known extensions of Stirling numbers due to Riordan, Carlitz, Howard, Charalambides-Koutras, Gould-Hopper, Hsu-Shiue, Tsylova Todorov, Ahuja-Enneking, and Stirling functions introduced by Butzer and Hauss, Butzer, Kilbas, and Trujilloet and others are included as particular cases of our generalization. Some basic properties related to our general pattern such as their recursive relations and generating functions are discussed. Three algorithms for calculating the Stirling numbers based on our generalization are also given, which include a comprehensive algorithm using the characterization of Riordan arrays.

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