1.1. Generai properties 9 Observe that by looking at the Wronskian of two solutions u G V±(\), v £ V±(X) it is not hard to see that the lemma becomes false if a(n) is exponentially decreasing. For later use observe that 0 - 1 1 0 (1.44) Sfon,™)- 1 = ^\j$(z,n,m)TJ-\ J a{m) (where 3T denotes the transposed matrix of £) and hence (1.45) |o(m)|||$(z,n,m)- 1 || = |a(n)|||*(z,n,ro)||. We will exploit this notation later in this monograph but for the moment we return to our original point of view. The equation (1.46) (r -z)f = g for fixed z G C, g G -^(Z), is referred to as inhomogeneous Jacobi equation. Its solution can can be completely reduced to the solution of the corresponding homogeneous Jacobi equation (1.19) as follows. Introduce ( L 4 7 ) , ,_a(z,n,m) Then the sequence (1.48) where K(z,n,m)= . a[m) f(n) = foc(z,n,n0) + fis{z,n,n0) n + ^ * K(z,n,m)g(m), m~no-{-l n-l (1.49) E * f0)={ J=n0 ]P fU) for n n o 0 for n = no n0 —1 - ^2 f(fi for n n ° j=n satisfies (1.46) and the initial conditions /(no) = /o, /(^o + 1) = /i a s c a n De checked directly. The summation kernel K(z,n,m) has the following properties: K(z, n, n) = 0, if (z, n -f 1, n) = a(n)~1, K(z, n, m) = —K(z, m, n), and ^(z, m)i(z, n) u(z, n)f (z, m) (1.50) K(z,n, m) W(u(z),v(z)) for any pair i^z), ^(Z) of linearly independent solutions of ru = z?x. Another useful result is the variation of constants formula. It says that if one solution u of (1.19) with u(n) ^ 0 for all n G Z is known, then a second (linearly independent, W(u,v) = 1) solution of (1.19) is given by n - l (1.51) v(n) =u(n) Y2 3=no aU)u(j)u{j + l)'
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