Definition 8.8.
For all real numbers \(p\) and nonnegative integers \(k\text{,}\) the number \(P(p,k)\) is defined by
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\(P(p,0)=1\) for all real numbers \(p\) and
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\(P(p,k) = p P(p-1,k-1)\) for all real numbers \(p\) and integers \(k>0\text{.}\)
Section 8.3 Newton’s Binomial Theorem
In Chapter 2, we discussed the binomial theorem and saw that the following formula holds for all integers \(p\ge1\text{:}\)
\begin{equation*}
(1+x)^p =\sum_{n=0}^{p}\binom{p}{n} x^n.
\end{equation*}
You should quickly realize that this formula implies that the generating function for the number of \(n\)-element subsets of a \(p\)-element set is \((1+x)^p\text{.}\) The topic of generating functions is what leads us to consider what happens if we encounter \((1+x)^p\) as a generating function with \(p\) not a positive integer. It turns out that, by suitably extending the definition of the binomial coefficients to real numbers, we can also extend the binomial theorem in a manner originally discovered by Sir Isaac Newton.
We’ve seen several expressions that can be used to calculate the binomial coefficients, but in order to extend \(C(p,k)\) to real values of \(p\text{,}\) we will utilize the form
\begin{equation*}
\binom{p}{k}=\frac{P(p,k)}{k!},
\end{equation*}
recalling that we’ve defined \(P(p,k)\) recursively as \(P(p,0)=1\) for all integers \(p\geq 0\) and \(P(p,k)=p P(p-1,k-1)\) when \(p\geq k > 0\) (\(k\) an integer). Notice here, however, that the expression for \(P(p,k)\) makes sense for any real number \(p\text{,}\) so long as \(k\) is a non-negative integer. We make this definition formal.
(Notice that this definition does not require \(p\geq k\) as we did with integers.)
We are now prepared to extend the definition of binomial coefficient so that \(C(p,k)\) is defined for all real \(p\) and nonnegative integer values of \(k\text{.}\) We do this as follows.
Definition 8.9.
For all real numbers \(p\) and nonnegative integers \(k\text{,}\)
\begin{equation*}
\binom{p}{k}=\frac{P(p,k)}{k!}.
\end{equation*}
Note that \(P(p,k)=C(p,k)= 0\) when \(p\) and \(k\) are integers with \(0\le p\lt k\text{.}\) On the other hand, we have interesting new concepts such as \(P(-5,4)=(-5)(-6)(-7)(-8)\) and
\begin{equation*}
\binom{-7/2}{5}=\frac{(-7/2)(-9/2)(-11/2)(-13/2)(-15/2)}{5!}.
\end{equation*}
With this more general definition of binomial coefficients in hand, we’re ready to state Newton’s Binomial Theorem for all non-zero real numbers. The proof of this theorem can be found in most advanced calculus books.
Theorem 8.10. Newton’s Binomial Theorem.
For all real \(p\) with \(p\neq0\text{,}\)
\begin{equation*}
(1+x)^p=\sum_{n=0}^\infty\binom{p}{n}x^n.
\end{equation*}
Note that the general form reduces to the original version of the binomial theorem when \(p\) is a positive integer.
