content_image/365259.jpg content_image/365258.jpg content_image/365257.jpg content_image/365256.jpg content_image/365255.jpg content_image/365254.jpg content_image/365253.jpg content_image/365252.jpg
and then obviously 

The q -binomial can now be defined in an alternative way as 

and then it follows that 

Quite analogously it is easy to verify that 

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<img src='content_image/365258.jpg'>

<img src='content_image/365257.jpg'>

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Finally, note also that [ n k ] q can be viewed as a formal polynomial in q of degree k ( n − k ) where the coefficient of q j counts the number of k -subsets of { 1 , . . . , n } with element sum j + k ( k + 1) / 2. It is thus a polynomial with positive coefficients. 

We have so far only stated what belongs to the standard repertoire on the subject. The q -binomials and the q -Pochhammer function have many interesting properties and we point the interested reader to the books $^{1}$,$^{2}$,3 and especially the charming little book $^{4}$. For more on the standard binomial coefficient and Pochhammer function we recommend 5 which contains a wealth of useful information. 

A natural extension of the q -binomial coefficient, the p, q -binomial coefficient, was defined in 6 as k 

Clearly, in the case p = 1 this reduces to a q -binomial coefficient. Also, note that p and q are interchangable so that 

<img src='content_image/365255.jpg'>

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Just as the standard binomial coefficients, their p, q -analogues are also symmetric 

<img src='content_image/365253.jpg'>

It is an easy exercise to show the following identity and we leave this to the reader. 

<img src='content_image/365252.jpg'>

