$^{GLYPH<Minus>}$1.5 $^{GLYPH<Minus>}$1.0 $^{GLYPH<Minus>}$0.5 0.5 1.0 1.5 x 0.2 0.4 0.6 0.8 1.0 R FIG. 1: R$_{p,q}$ ( n, n/ 2 , x n 3 / $^{4}$) (blue curves) versus x at z = − 9 and y = 2 w = − 0 . 105391 for n = 2 $^{6}$, 2 $^{8}$, 210 and 2 $^{12}$. The red curve is the asymptote e − 0 . 719718 x $^{4}$. This puts the maximum at x = ± √$_{a}$ for a > 0 and at x = 0 for a ≤ 0. Figure 2 works like figure 1 but here with the parameter a set to 1. It shows R$_{p,q}$ ( n, n/ 2 , x n 3 / $^{4}$) for z = − 9 with p = 1 − 0 . 105391 /n − 1 . 50171 /n 3 / 2 and the asymptotic ratio is given by (65), that is, e − 0 . 719718 ( x $^{4}$− 2 x $^{$^{2}$)}$. The red curve is the asymptote and the blue curves are for finite n where the larger n are closer to the asymptote. Had we set a < 0 the distributions would still be wide but with a single peak in the middle. If we instead choose a middle ratio of 1+ a/n then we receive a sharply peaked distribution. Say that we want C. Peakish distributions R$_{p,q}$ ( n, n/ 2 , 1) = p n/ 2 − q n/ 2 p n/ 2+1 q n/ 2+1 = 1 + a n (66) 20