<b>A Note on the Generalized Gamma Function</b> In this paper, the generalized gamma functions of the first and second types are firstly introduced and investigated. It can be proven that the traditional gamma function is a special case of the first type of generalized gamma function. Besides, the iterative formula of the generalized gamma function will be fully derived. Finally, a numerical example is provided to illustrate the validity and effectiveness of our main result. Generalized gamma function, gamma function, iterative formula, factorial function 1502-1504 Issue-1 Volume-5 Yeong-Jeu Sun