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

Abstract: We derive product and series representations of the gamma function using Newton interpolation series. Using these identities, a new formula for the coefficients in the Taylor series of the reciprocal gamma function is found. We also find two new series representations 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 is convergent for the entire real axis. Finally, we conjecture a novel series representation for the principal branch of the inverse gamma function $\text{inv}\Gamma_0(z)$.