Concept of Factorials and Combinatorial Numbers and its Consequences for Algebra and Analysis

In this article, the usual factorials and binomial coefficients have been generalized and extended to negative integers. Based on this generalization and extension, a new kind of polynomials has been proposed, which has directly led to the non-classical hypergeometric orthogonal polynomials and the non-classical second-order hypergeometric linear ordinary differential equations. The resulting polynomials can be used in non-relativistic and relativistic quantum mechanics, particularly in the case of the Schrödinger equation and Dirac equations for an electron in a Coulomb potential field.