Moments of {x„} The purpose of this subsection is to see if something can be said about the ratio (Ekkx,)/(1k X0 when {xjn, is a non-zero solution to the system 0 = -q) n,1 Oarx - (TS, + di)x, 0 = q San., xn.1 - (qta,, + d„) x„. (1.28) with a suitable constant. To this end, introduce by way of notation; = I ,,A)a„x„. The equations in (1.28) can be used to derive two expressions for ;, these being • Xn = oisani+ do (110.iska (goatir+k do )Citaj Xj for n 2. • xy, — mani+ do (niska (qoacitak+kdk) )(1 q)S. (1.29) Note that 4) must be such that _ q v oak - 1 . q k=1 (Oak +dk ) (1.30) This last condition can be restated as saying that and therefore This tells us that ins2(qOan+dn)x• = cic- q; - (with, +di); + En.g, d„ x„ = q; E„,d„x„ =(qtal+dax, = (1 -q); , (1.31) (1.32) (1.33) where the left hand inequality comes via the n = 1 version of (1.29). What is written in (1.33) is of at least two identities involving `moments' of @ca. To elaborate, introduce a variable t and use (1.29) to see the equality between the following two formal series: ((clOan+d,,)x.) = En, t° (94)a „x,,) (1.35) Let Q(t) denote the series En, ta(qta„x„) and let p(t) denote En, Then (1.35) says that EFTA01155621