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Lemma IV.8. Let q = n + a n +2 where a > 0 . Then 

<img src='content_image/352440.jpg'>

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. 

Lemma IV.9. Let q = n + a n +2 and a > 0 . Then 

Theorem IV.10. Let q = n + a n +2 where a > 0 . Then 

<img src='content_image/352437.jpg'>

<img src='content_image/352436.jpg'>

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. 

<img src='content_image/352435.jpg'>

<img src='content_image/352432.jpg'>

<img src='content_image/352433.jpg'>

<img src='content_image/352434.jpg'>

