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## Theorem II.1. 

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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 . 

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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 → ∞ . 

