Another way to derive this same result is to start with Ck B, Ck-1 or Ck = Aß, (8.20) in equation (8.9). Substituting this function in equation (8.9) and simplifying gives qDe+1 – BDe + (pß²)De-1 = 0,...

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Extracted text: Another way to derive this same result is to start with Ck B, Ck-1 or Ck = Aß, (8.20) in equation (8.9). Substituting this function in equation (8.9) and simplifying gives qDe+1 – BDe + (pß²)De-1 = 0, (8.21) which has the characteristic equation qr2 – Br + pg? = 0. (8.22) Therefore, ri = B, PB r2 = (8.23) and De = A3(B)B + A4(B) (8.24) Since z(k, l, B) = CkDe, we have z(k, l, 3) = Ā3(3),3* +€+ Ā4(18) () 3*ie, (8.25) where [A3(B), A4(B), A3 (B), A4(3)] are arbitrary functions of B. If we now sum/integrate over ß, then z(k, l) = g1(k + l) + h1(k + l), (8.26) and h1 are functions of (k +l). Let's now show that equations (8.19) and (8.26) are the same. To do this, where 9i define a new discrete variable m = k + l. (8.27) Therefore, k m-l (:)" (;) (8.28) and k m h(k +l) = h(k + l) k+l h(k +l) h1(k + l). (8.29) This shows that equations (8.19) and (8.26) are equal.