Extending the binomial coefficients to preserve symmetry and pattern

AbstractWe show how to extend the domain of thee binomial coefficients (rn) so that n and r may take any integer value. We argue from two directions; on the one hand we wish to preserve symmetry and pattern within Pascal's triangle (thus, creating Pascal's hexagon), and on the other hand we wish the binomial coefficients to preserve their algebraic role in terms of the Taylor series and Laurent series expansions of (1 + x)n, valid when |x| < 1, |x| > 1, respectively.A geometric configuration within the Pascal triangle, called the Pascal flower, has some extraordinary properties—these properties persist into the hexagon. Moreover, the binomial coefficients may, by the use of the Γ-function, even be extended to all real (or complex) values of n and r, with the conservation of their principal properties