Lesson 3

The formula \[\binom{\alpha}{n}=\frac{\alpha(\alpha-1)\dotsb(\alpha-n+1)}{n!}\] generalizes the formula \[\binom{n}{k}=\frac{n(n-1)\dotsb (n-k+1)}{k!}\] for binomial coefficients to the case of an arbitrary real \(n\). (To emphasize that \(n\) is no longer a non-negative integer, we use \(\alpha\) instead of \(n\). The parameter \(k\) is replaced by \(n\), as throughout this chapter \(n\) denotes the required index in the sequence. We hope this sudden change of notation does not confuse readers too much.) The combinatorial meaning is lost, but various useful properties remain. Below, we will look at some examples that are also important special cases of the formula \((1+x)^{\alpha}=\sum_{n=0}^{\infty}\binom{\alpha}{n}x^{n}\).