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 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 Clearly, in the case p = 1 this reduces to a q -binomial coefficient. Also, note that p and q are interchangable so that Just as the standard binomial coefficients, their p, q -analogues are also symmetric It is an easy exercise to show the following identity and we leave this to the reader. 5 [ n k ] p,q = p k ( $^{n}$−k ) [ n k ] q/p = q k ( $^{n}$−k ) [ n k ] p/q (17) [ n k ] p,q = [ n n − k ] p,q (16) [ n k ] p,q = [ n k ] q,p (15) [ n k ] p,q = k ∏ i =1 p $^{n}$−i +1 − q $^{n}$−i +1 p i q i , p = q , 0 ≤ k ≤ n (14) lim $^{q}$→ 1 ( q $^{a}$; q )$_{n}$ (1 − q ) n = ( a )$_{n}$ (13) lim $^{q}$→ 1 [ n k ] q = ( n k ) (12) [ n k ] q = [ n ]$_{q}$! [ k ]$_{q}$! [ n − k ]$_{q}$! (11) lim $^{q}$→ $_{1}$[ n ]$_{q}$! = n ! (10)