A Unified Approach to Generalized Stirling Functions

中文关键词:  Stirling numbers  Stirling functions  factorial polynomials  generalized factorial  divided difference  $k$-Gamma functions  Pochhammer symbol and $k$-Pochhammer symbol.

英文关键词:Stirling numbers  Stirling functions  factorial polynomials  generalized factorial  divided difference  $k$-Gamma functions  Pochhammer symbol and $k$-Pochhammer symbol.

中文摘要:

      Here presented is a unified approach to generalized Stirling functions by using generalized factorial functions, $k$-Gamma functions, generalized divided difference, and the unified expression of Stirling numbers defined in [16]. Previous well-known Stirling functions introduced by Butzer and Hauss [4], Butzer, Kilbas, and Trujilloet [6] and others are included as particular cases of our generalization. Some basic properties related to our general pattern such as their recursive relations, generating functions, and asymptotic properties are discussed, which extend the corresponding results about the Stirling numbers shown in [21] to the defined Stirling functions.

英文摘要:

      Here presented is a unified approach to generalized Stirling functions by using generalized factorial functions, $k$-Gamma functions, generalized divided difference, and the unified expression of Stirling numbers defined in [16]. Previous well-known Stirling functions introduced by Butzer and Hauss [4], Butzer, Kilbas, and Trujilloet [6] and others are included as particular cases of our generalization. Some basic properties related to our general pattern such as their recursive relations, generating functions, and asymptotic properties are discussed, which extend the corresponding results about the Stirling numbers shown in [21] to the defined Stirling functions.