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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