Theorem II.1. [ m + n k ] p,q = k ∑ ℓ =0 [ m k − ℓ ] p,q [ n ℓ ] p,q p ( $^{n}$−ℓ ) ( $^{k}$−ℓ $^{)}$q ℓ ( $^{m}$−k + ℓ ) (26) Proof. In the q -Vandermonde identity (25), replace q with q/p and multiply both sides with p k ( m + $^{n}$−k $^{)}$. Using (17) the left hand side is now a pure p, q -binomial coefficient. The ℓ th term of the right hand side is and the theorem follows. The q -binomial coefficients have been shown to form a log-concave (and thus unimodal) sequence for q ≥ 0, see e.g. 7 and $^{8}$. However, for the p, q -binomial coefficients this does not always hold. Rewriting them as a product like in (17) we have in fact a product of two sequences; that of p k ( $^{n}$−k ) and [ n k ] q/p for k = 0 , . . . , n . The first sequence is log-concave for p ≥ 1 and log-convex for p ≤ 1. It is well-known that the element-wise product of two log-concave positive sequences is also log-concave. So, if p ≥ 1 and q ≥ 0 then the sequence of [ n k ] p,q is log-concave. We conjecture that for p, q > 0 the sequence can be either unimodal, with the maximum at k = ⌊ n/ 2 ⌋ , or bimodal, with the maxima at k and n − k for some 0 ≤ k ≤ n/ 2, but not trimodal etc. We will assume this to be true in this paper but a formal proof is still lacking. Note that if we allow negative values of p the sequence can have a local maximum at every alternate index k . We write f ( n ) ∼ g ( n ) to denote that f ( n ) /g ( n ) → 1 as n → ∞ . Analogously f ( n ) ∝ g ( n ) denotes that f ( n ) /g ( n ) → A , for some non-zero real number A , as n → ∞ . 7 [ m k − ℓ ] q/p [ n ℓ ] q/p p k ( m + $^{n}$−k $^{)}$( q/p ) ℓ ( $^{m}$−k + ℓ ) = [ m k − ℓ ] q/p [ n ℓ ] q/p p k ( m + $^{n}$−k $^{)}$− ℓ ( $^{m}$−k + ℓ $^{)}$q ℓ ( $^{m}$−k + ℓ ) = p ( $^{k}$−ℓ ) ( $^{m}$− ( $^{k}$−ℓ )) [ m k − ℓ ] q/p p ℓ ( $^{n}$−ℓ ) [ n ℓ ] q/p p ( $^{n}$−ℓ ) ( $^{k}$−ℓ $^{)}$q ℓ ( $^{m}$−k + ℓ ) = [ m k − ℓ ] p,q [ n ℓ ] p,q p ( $^{n}$−ℓ ) ( $^{k}$−ℓ $^{)}$q ℓ ( $^{m}$−k + ℓ )