# Section8.4An Application of the Binomial Theorem¶ permalink

In this section, we see how Newton's Binomial Theorem can be used to derive another useful identity. We begin by establishing a different recursive formula for $P(p,k)$ than was used in our definition of it.

Our goal in this section will be to invoke Newton's Binomial Theorem with the exponent $p=-1/2\text{.}$ To do so in a meaningful manner, we need a simplified expression for $C(-1/2,k)\text{,}$ which the next lemma provides.

##### Proof

Now recalling Proposition 8.3 about the coefficients in the product of two generating functions, we are able to deduce the following corollary of Theorem 8.12 by squaring the function $f(x) = (1-4x)^{-1/2}\text{.}$