Series and Product Representations of Gamma, Pseudogamma and Inverse Gamma Functions

We derive and prove product and series representations of the gamma function using Newton interpolation. We then show how these equations can be used to construct better and better approximations to the gamma function by writing it as a product over the prime numbers. The series definition is also used to find a new representation for the Euler-Mascheroni constant, containing only rational terms. After that, we introduce a new pseudogamma function which we call the Lambda function. This function interpolates the factorial at the positive integers, the reciprocal factorial at the negative integers and converges for the entire real axis. Finally, we conjecture a novel series representation for the principal branch of the inverse gamma function Γ(y)=x.