III. THE p, q -BINOMIAL DISTRIBUTION Now we are ready to introduce the notation and define the p, q -binomial probability function The reader should here observe that the sum of the coefficients has, to the best of our knowledge, no simpler expression in the general case. Neither do the sum of the q -binomial coefficients have a simpler expression that we are aware of. Compare this with the case of the standard binomial coefficients for which the sum is simply 2 $^{n}$. Having made our assumption of unimodality/bimodality we can now set up a simple computational scheme to find values of p given q . First we need to define a highly useful quantity; the ratio between two coefficients In the special case when ℓ = 1 we are looking at two consecutive coefficients. The ratio then becomes Lemma III.1. Let k, n be positive integers such that 1 ≤ k ≤ n − k . If x > 1 then 8 Ψ$_{p,q}$ ( n ) = n ∑ k =0 [ n k ] p,q (27) P$_{p,q}$ ( n, k ) = [ n k ] p,q Ψ$_{p,q}$ ( n ) (28) R$_{p,q}$ ( n, k, ℓ ) = [ n k − ℓ ] p,q [ n k ] p,q , 0 ≤ ℓ ≤ k ≤ n/ 2 (29) R$_{p,q}$ ( n, k, 1) = [ n k − 1 ] p,q [ n k ] p,q = p k − q k p $^{n}$−k +1 q $^{n}$−k +1 = p − ( $^{n}$− 2 k +1) 1 − ( q/p ) k 1 − ( q/p ) $^{n}$−k +1 = q − ( $^{n}$− 2 k +1) 1 − ( p/q ) k 1 − ( p/q ) $^{n}$−k +1 (30) 0 < 1 − x k 1 − x $^{n}$−k +1 < k n − k + 1