12 1. Jacobi operators Our next task will be expansions of c(z, n, no), s{z, n, no) for large z. Let Jni,n2 be the Jacobi matrix / b(rn + 1) a(ni + 1) a(m + 1) 6(m + 2) (1.64) Jni,n2 fc(n2 - 2) a(n2 - 2) a ( n 2 - 2 ) b(n2-l) ) Then we have the following expansion for s(z,n, no), n no, (1.65) s(z,n,n0) = det(z - Jn^n) ^ - E j = lPn0,n(j k-j n"=n0 + l 3) Uj = la(n0+j) where k = n no 1 0 and (1.66) Pno,nU) tr(J4,n)-E =JPn0lnWtr(J4-l) x . fc To verify the first equation, use that if z is a zero of s(.,n,no), then (s(z,no + 1, no),..., s(z, n 1, no)) is an eigenvector of (1.64) corresponding to the eigenvalue z. Since the converse statement is also true, the polynomials (in z) s(z,n, no) and det(z Jno,n) only differ by a constant which can be deduced from (1.30). The second is a well-known property of characteristic polynomials (cf., e.g., [91]). The first few traces are given by n0 + k tr(J no , no+fc+ i) = ] T 6(j), no+fc no+fc —1 tr(J20,no+fc+1) = £ b(j)2 + 2 £ a(j)2, j = n 0 + l j ' = n 0 + l no+fc no+fc —1 tr(J^, n o + f e + 1 ) = £ 6(i)3 + 3 £ a(i)2(6(j) + b(j + 1)), no+fc no + fc 1 tr(J^„ 0 + f e + 1 ) = £ 6 W 4 - 4 j a(j-)2(6(j)2 + K i + 1 W ) j ' = n 0 + l (1.67) 2 n 0 +/c —2 +Hi+i) 2 +^)+ 4 x «o'+i)2- j = n 0 + l
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