content_image/362067.jpg content_image/362068.jpg content_image/362069.jpg content_image/362070.jpg
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 

<img src='content_image/362067.jpg'>

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). 

## VI. CONTROLLING THE p, q -BINOMIAL DISTRIBUTION 

<img src='content_image/362068.jpg'>

<img src='content_image/362069.jpg'>

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. 

## A. Wide and flat distributions 

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 

<img src='content_image/362070.jpg'>

