Note that above, when the coefficients had a rather wide distribution, we examined their behaviour at x n 3 / 4 from the middle. Under our current assumption of q the distribution gets more sharply concentrated around the middle (basically they become gaussian), thus we study their behaviour at $_{x}$√$_{n}$ from the middle. Theorem IV.10. Let q = n + a n +2 where a > 0 . Then If we decrease a below zero the sequence becomes sharply bimodal, with all its mass concentrated around two peaks. We can of course connect the position of the peaks with the parameter a . Suppose that we want one of the peaks to have its maximum located at k and k − 1, that is, let the ratio here be 1. Lemma IV.11. Given a number µ such 0 < | µ | < 1 let If then Proof. A simple calculation shows that the limit of the ratio is Setting the limit to 1 and solving the equation gives the lemma. 14 k = n 2 (1 + µ ) a = 2 ( 1 − atanh µ µ ) lim $^{n}$→∞ R$_{q,q}$ ( n, k, 1) = 1 lim $^{n}$→∞ R$_{q,q}$ ( n, k, 1) = 1 + µ 1 − µ exp ( µ ( a − 2)) Ψ$_{q,q}$ ( n ) ∼ $^{√$_{n}$}$[ n n/ 2 ] q,q $^{+}$∞ ∫ −∞ exp ( − a x $^{2}$) d x ∼ √ 2 e a ( 2 √$_{e}$ ) n exp ( a n 4 − a 2 8 ) R$_{q,q}$ ( n, n/ 2 $^{$_{, x}$√$_{n}$}$) ∼ exp ( − a x $^{2}$) Lemma IV.9. Let q = n + a n +2 and a > 0 . Then Lemma IV.8. Let q = n + a n +2 where a > 0 . Then [ n n/ 2 ] q,q ∼ √ 2 e π n ( 2 √$_{e}$ ) n exp ( a n 4 − a 2 8 )