content_image/331780.jpg content_image/331781.jpg content_image/331782.jpg content_image/331783.jpg content_image/331784.jpg content_image/331785.jpg content_image/331786.jpg content_image/331787.jpg
As a corollary it follows that 

Any identity involving q -binomial coefficients can then be extended to a p, q -binomial identity by first replacing q with q/p and then use the identity (17). For example, the binomial theorem n 

has the q -analogue 

Now replace q with q/p and then use the identity 

from (17) above. This gives us 

<img src='content_image/331780.jpg'>

<img src='content_image/331781.jpg'>

<img src='content_image/331782.jpg'>

<img src='content_image/331783.jpg'>

<img src='content_image/331784.jpg'>

After multiplying both sides with p ( n $_{2}$) this simplifies into 

which was also shown in 6 using a recursion technique. 

As another application we consider the Chu-Vandermonde identity 

which is a fairly direct consequence of the binomial theorem applied to the product (1 + x ) $^{m}$(1+ x ) $^{n}$. The q -Vandermonde identity, see e.g. 1 and (though misprinted) $^{6}$, can be stated as k 

Using (17) above we can now obtain a p, q -analog of this. 

<img src='content_image/331785.jpg'>

<img src='content_image/331786.jpg'>

<img src='content_image/331787.jpg'>

