Moments of (x„) The purpose of this subsection is to see if something can be said about the ratio (I kk xk)/(lk xk) when {xk}kz, is a non-zero solution to the system 0 = (1 -q)Ezz, $a„X, - (q4)a, + di)x, 0 = qta„.lx„., - (q4)a„ + di,)x„. (1.28) with 4) a suitable constant. To this end, introduce by way of notation; = Ezz,4)a„x„. The equations in (1.28) can be used to derive two expressions for x„, these being • Oak X. = (q*ant+ do) (ILIA°, (oak +do )qtai xi for n 2 2. • Oak X^ = (04 + 4) In ist<" (qOak + 4) v n i nmr•, • (1.29) Note that 4) must be such that 1 . I q Lie wk., (oak +dk) = (1.30) This last condition can be restated as saying that and therefore This tells us that 1„,2(q4);+d„); = q; q; - (q41a,+d,)x, + Enz,(1,,xn = (N. f ,,z, clax„ = (q4)a, +di) x1= (I -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 {x,}. To elaborate, introduce a variable t and use (1.29) to see the equality between the following two formal series: E„,2 t" ((q4)a„+ d„) x„) = Z„,z, t°(q4m,,x„) (1.35) Let Q(t) denote the series Ent, tn(q4ia„x„) and let go(t) denote I,, in d.x,,. Then (1.35) says that EFTA_R 1 _02 009 1 54 EFTA02685614