EFTA02685614.pdf

Moments of (x„) The purpose of this subsection is to see if something can be said about the ratio (I k k 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 ni m , • nr• (1.29) Note that 4) must be such that I q Lie wk.,(oak+dk)= 1 . (1.30) This last condition can be restated as saying that 1„,2 (q4);+d„); = q; (1.31) and therefore q; - (q41a,+d,)x, + Enz,(1,,xn = (N. (1.32) This tells us that f ,,z, cla x„ = (q4)a, +di) x1= (I -q); , (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_R1_02009 1 54 EFTA02685614  r' (2(t) + r' p(t) = Q(t)+ (qtal +di) xi (1.36) This in turn can be rewritten using (1.33) as p(t) =(t- II) Q(0+ t (1-q); (1.37) Taking t = 1 on both sides recovers (1.33): E„,, d„x„ = (I -q)c. Differentiating once and setting t = I finds Ew nd„xn = Q(1)+(l -q);. (1.38) To go further, use (1.31) to see that = -E.I d„x„ + qc+(qtlmi +di)x, =q; (1.39) Granted this last equality, then (1.38) asserts that En ind„x„=;. (1.40) This with (1.33) says that Zia] nd.x. L ai do. — (1-4) (1.41) In the case d„ = d for all n, this asserts what is conjectured by Martin. Identities for `moments' of the form L a, nPd„x„ for p a 2 require knowing something of the (p-l)'st derivative of Q at t = 1. I don't know any good way to obtain these. EFTA_R1_02009155 EFTA02685615