Binomial Coefficients As Polynomials
For any nonnegative integer k, the expression can be simplified and defined as a polynomial divided by k!:
This presents a polynomial in t with rational coefficients.
As such, it can be evaluated at any real or complex number t to define binomial coefficients with such first arguments. These "generalized binomial coefficients" appear in Newton's generalized binomial theorem.
For each k, the polynomial can be characterized as the unique degree k polynomial p(t) satisfying p(0) = p(1) = ... = p(k − 1) = 0 and p(k) = 1.
Its coefficients are expressible in terms of Stirling numbers of the first kind, by definition of the latter:
The derivative of can be calculated by logarithmic differentiation:
Read more about this topic: Binomial Coefficient