A Unified Generalization of the Catalan, Fuss, and Fuss–Catalan Numbers

Abstract

In the paper, the authors introduce a unified generalization of the Catalan numbers, the Fuss numbers, the Fuss−Catalan numbers, and the Catalan−Qi function, and discover some properties of the unified generalization, including a product-ratio expression of the unified generalization in terms of the Catalan−Qi functions, three integral representations of the unified generalization, and the logarithmically complete monotonicity of the second order for a special case of the unified generalization.

Keywords

• unified generalization
• Catalan number
• Fuss number
• Fuss–Catalan number
• Catalan–Qi function
• product-ratio expression
• logarithmically complete monotonicity of the second order
• integral representation