A new definition of generalization of the factorial function with new results about the gamma of negative integers

Abstract

Abstract The gamma function is a mathematical function that generalizes the concept of factorial to real and complex numbers. While the gamma function is a powerful tool in mathematics, it does have certain limitations and potential issues for example, · Non-integer values: The gamma function is not defined for negative integers, this limitation can be problematic in certain contexts where negative integer values are involved. · Pole at zero: The gamma function has a pole at zero, which means it is undefined at this point. This can pose challenges when working with functions or equations that involve the gamma function near or at zero. · Computational complexity: Computing the gamma function numerically can be computationally expensive and time-consuming, especially for large or complex arguments. In this study, we have addressed the aforementioned issues by proposing a new definition for generalizing the factorial function, which serves as an alternative definition of the gamma function. This new definition is formulated based on the utilization of the differential operator. The resulting definition stands out for its simplicity and effectiveness in computing real numbers, including non-positive integers. Moreover, our research has yielded fresh insights into the gamma function's behavior with respect to non-positive integers, potentially leading to a transformative approach in employing fractional differential and integral equations to describe a wide range of cosmic phenomena.