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 has the q -analogue Now replace q with q/p and then use the identity from (17) above. This gives us 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 Using (17) above we can now obtain a p, q -analog of this. 6 lim $^{p,q}$→r [ n k ] p,q = r k ( $^{n}$−k ) ( n k ) (18) (1 + x ) n = n ∑ k =0 ( n k ) x k (19) $^{n}$− 1 ∏ ℓ =0 ( 1 + x q $^{ℓ}$) = n ∑ k =0 [ n k ] q q ( k $_{2}$) x k (20) [ n k ] q/p = p − k ( $^{n}$−k ) [ n k ] p,q (21) $^{n}$− 1 ∏ ℓ =0 ( 1 + x ( q/p ) $^{ℓ}$) = n ∑ k =0 [ n k ] p,q p − k ( $^{n}$−k $^{)}$( q/p ) ( k $_{2}$) x k (22) $^{n}$− 1 ∏ ℓ =0 ( p ℓ + x q $^{ℓ}$) = n ∑ k =0 [ n k ] p,q p ( n − k 2 ) q ( k $_{2}$) x k (23) ( m + n k ) = k ∑ ℓ =0 ( m k − l ) ( n ℓ ) (24) [ m + n k ] q = k ∑ ℓ =0 [ m k − ℓ ] q [ n ℓ ] q q ℓ ( $^{m}$−k + ℓ ) (25)