The binomial coefficient \(C\binom{n}{x}\) for arbitrary \(x\)

Binomial coefficients of the form \( \binom{\alpha}{\beta} \) for complex numbers \( \alpha \) and \( \beta \) can be defined in terms of the gamma function, or equivalently the generalized factorial function. Less well-known is the fact that if \( n \) is a natural number, the binomial coefficient \( \binom{n}{x} \) can be defined in terms of elementary functions. This enables us to investigate the function \( \binom{n}{x} \) of the real variable \( x \). The results are completely in line with what one would expect after glancing at the graph of \( \binom{3}{x} \), for example, but the techniques involved in the investigation are not the standard methods of calculus. The analysis is complicated by the existence of removable singularities at all of the integer points in the interval \( [0, n] \), and requires multiplying, rearranging, and differentiating infinite series.