Using the bounds for the q -Pochhammer function we can now bound the ratio. For the upper bound of the ratio we take the quotient of the upper bound and the lower bound. The ratio R$_{q}$ ( n, k, ℓ ) then has the upper bound and, quite analogously, it has the lower bound Using (17) we can now obtain bounds for the quotients of p, q -binomial coefficients. Note simply that and use the bounds from (49) and (50). Choosing a k for a given n such that R$_{p,q}$ ( n, k, 1) = 1 defines a set of pairs p, q ; an isocurve. If we choose a value of q , then what value should p have to result in a distribution of coefficients which has a peak at k (and n − k ), i.e. with R$_{p,q}$ ( n, k, 1) = 1? The iterative method (31) above produces the correct p for any given q but reveals no information on p . To obtain this we need to parameterize q properly and one way to do this is to set q = 1+ z/n for some z ≤ 0. This parameterization was also used in 9 and 10 for computing giving upper bounds on q -binomial coefficients. Let us begin with the particular distribution which has its peak in the middle, i.e. k = n/ 2. Due to symmetry we are interested only in the case p > q . Recall that in the special case p = q the distribution has its peak at n/ 2 when A. Wide and flat distributions VI. CONTROLLING THE p, q -BINOMIAL DISTRIBUTION R$_{q}$ ( n, k, ℓ ) ≤ exp ( Li$_{2}$ ( q k − ℓ +1 ) +Li$_{2}$ ( q n − k + ℓ ) − Li$_{2}$ ( q k +1 ) − Li$_{2}$ ( q n − k +1 ) log q ) 1 − q $^{n}$−k +1 (49) R$_{q}$ ( n, k, ℓ ) ≥ ( 1 − q $^{k}$−ℓ $^{+1}$) exp ( Li$_{2}$ ( q k − ℓ +1 ) +Li$_{2}$ ( q n − k + ℓ +1 ) − Li$_{2}$ ( q k ) − Li$_{2}$ ( q n − k +1 ) log q ) (50) R$_{p,q}$ ( n, k, ℓ ) = R$_{q/p}$ ( n, k, ℓ ) p ℓ ( $^{n}$− 2 k + ℓ ) (51) q = n n + 2 = 1 − 2 n + 4 n 2 − 8 n 3 + · · · (52) 17