meaning that the quotient between the left- and right-hand side goes to 1 as n → ∞ . The proof follows from an easy application of the identity and we leave it to the reader. A somewhat more involved application of (35) is the following lemma This allows us to give the exact order of the sum. Proof. The calculations goes as follows though we leave out some details. where the factor n 3 / 4 in front of the integral comes from the change of variables k = x n 3 / $^{4}$. This is the sum at just a single point where the distribution becomes flat in the middle region. We can do even better if we allow ourselves to move around in the vicinity of this point. 12 Ψ$_{q,q}$ ( n ) = [ n n/ 2 ] q,q n/ 2 ∑ k $^{=}$− n/ 2 [ n n/ 2 + k ] q,q [ n n/ 2 ] q,q ∼ n 3 / 4 [ n n/ 2 ] q,q $^{+}$∞ ∫ −∞ exp ( − 4 3 x 4 ) d x ∼ n 3 / 4 √ 2 e π n ( 2 √$_{e}$ ) n 3 1 / $^{4}$Γ( 1 / 4 ) $_{2}$√$_{2}$ = Γ( 1 / 4 ) 3 1 / $^{4}$n1 / 4 √$_{π}$ ( 2 √$_{e}$ ) $^{n}$− 1 Ψ$_{q,q}$ ( n ) ∼ Γ( 1 / 4 ) 3 1 / $^{4}$n1 / 4 √$_{π}$ ( 2 √$_{e}$ ) $^{n}$− 1 R$_{q,q}$ ( n, n/ 2 , x n 3 / $^{4}$) ∼ exp ( − 4 3 x 4 ) ( 1 + x n ) n = e x ( 1 − x 2 2 n + x 3 3 n 2 + x 4 8 n 2 + · · · ) (35) [ n n/ 2 ] q,q ∼ √ 2 e π n ( 2 √$_{e}$ ) n Lemma IV.2. For q = n n +2 we have Lemma IV.3. Let x be some real number. For q = n n +2 we have Theorem IV.4. For q = n n +2 we have