On the Formalization of Gamma Function in HOL

The Gamma function is a special transcendental function that is widely used in probability theory, fractional calculus and analytical number theory. This paper presents a higher-order logic formalization of the Gamma function using the HOL4 theorem prover. The contribution of this paper can be mainly divided into two parts. Firstly, we extend the existing integration theory of HOL4 by formalizing a variant of improper integrals using sequential limits. Secondly, we build upon these results to formalize the Gamma function and verify some of its main properties, such as pseudo-recurrence relation, functional equation and factorial generalization. In order to illustrate the practical effectiveness and utilization of our work, we formally verify some properties of Euler’s generalized power rule of differentiation, Mittag-Leffler functions and the relationship between the Exponential and Gamma random variables.

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