We continue as before and estimate the growth rate of the peak coefficient. The result (and the proof) is somewhat more complicated but follows from an application of (35). Again we take $_{x}$√$_{n}$ steps away from the peak and find the shape of the distribution. $_{Finally we get the sum by multiplying the integral with the peak coefficient and 2}$√$_{n}$$_{,}$ where the factor 2 is due to that we have two peaks. where the growth rate of the peak coefficient is that of lemma IV.12. In this section we set up bounds for the q -Pochhammer function and use them for giving bounds of quotients between q -binomial coefficients. Basically we mimic the upper bound in 9 and 10 but we extend it to obtain a lower bound as well. They are very useful bounds so we will do this in some detail though everything is based on standard elementary methods. First we need the integral estimate of a sum. Let f ( x ) be a continuous, positive, decreasing function on the interval m ≤ x ≤ n + 1 where m and n are integers. Then V. POCHHAMMER BOUNDS n +1 ∫ m f ( x ) d x ≤ n ∑ k = m f ( k ) ≤ f ( m ) + n ∫ m f ( x ) d x (38) 15 Ψ$_{q,q}$ ( n ) ∼ $^{$_{2}$√$_{n}$}$[ n n 2 (1 + µ ) ] q,q $^{+}$∞ ∫ −∞ exp { 2 x 2 ( 1 µ 2 − 1 + atanh µ µ )} d x ∼ $^{√$_{2}$$_{π n}$}$[ n n 2 (1 + µ ) ] $_{q,q}$√ µ (1 − µ $^{2}$) µ + ( µ 2 − 1) atanh µ Theorem IV.14. Let µ and a be defined as in lemma IV.11 and set q = n + a n $_{+2}$. Then Lemma IV.12. Let µ and a be defined as in lemma IV.11 and set q = n + a n $_{+2}$. Then Lemma IV.13. Let µ and a be defined as in lemma IV.11 and set q = n + a n $_{+2}$. Then [ n n 2 (1 + µ ) ] q,q ∼ $^{√$_{2 exp}$}${ n 2 ( log 4 1 − µ 2 − 1+ µ 2 µ atanh µ ) + 1 − µ 2 2 µ 2 (2 $^{µ}$− atanh µ ) atanh µ } √ π n (1 − µ $^{2}$) R$_{q,q}$ ( n, n 2 (1 + µ ) $^{$_{, x}$√$_{n}$}$) ∼ exp { 2 x 2 ( 1 µ 2 − 1 + atanh µ µ )}