for some real number a . Then This can be verified using (35) as can the following lemma. Lemma IV.6. Let a and x be real numbers. For q = n n +2 + a n 3 / 2 we have We now have the resources to estimate the sum of the coefficients for a whole spectrum of values of q near n/ ( n + 2). The next theorem can be shown using the same technique as Theorem IV.4, though the result gets slightly more complicated due to the integral on the right hand side in the previous lemma. For a < 0 the asymptotic order of Ψ$_{q,q}$ ( n ) is Here I$_{α}$ ( x ) and K$_{α}$ ( x ) denote the modified Bessel functions of the first and second kind respectively. The m th moment is simpler to express using an integral formulation. and the asymptotic behaviour of the middle coefficient is given by lemma IV.5. The same technique allows us to repeat this for points farther away from the critical point q = n/ ( n + 2). If we increase q by a/n then the coefficients get sharply concentrated in the middle like that of standard binomial coefficients. Again (35) to the rescue. 13 n/ 2 ∑ k $^{=}$− n/ $_{2}$| k | m [ n n 2 + k ] q,q ∼ (36) n 3 m +3 4 [ n n/ 2 ] q,q $^{+}$∞ ∫ −∞ | x | $^{m}$, exp ( − a x 2 − 4 3 x 4 ) d x (37) n 1 / 4 4 √ − 3 a e π ( 2 √$_{e}$ ) n exp ( $_{a}$√$_{n}$ 4 + 3 a 2 32 ) ( I$_{1}$$_{/}$$_{4}$ ( 3 a 2 32 ) + I − 1 / $_{4}$( 3 a 2 32 ) ) n 1 / 4 4 √ 6 a e π ( 2 √$_{e}$ ) n exp ( $_{a}$√$_{n}$ 4 + 3 a 2 32 ) K$_{1}$$_{/}$$_{4}$ ( 3 a 2 32 ) R$_{q,q}$ ( n, n/ 2 , x n 3 / $^{4}$) ∼ exp ( − a x 2 − 4 3 x 4 ) [ n n/ 2 ] q,q ∼ √ 2 e π n ( 2 √$_{e}$ ) n exp ( $_{a}$√$_{n}$ 4 ) q = n n + 2 + a n 3 / 2 Lemma IV.5. Let Theorem IV.7. Let q = n n +2 + a n 3 / $_{2}$. For a > 0 the asymptotic order of Ψ$_{q,q}$ ( n ) is