The paper deals with the spectral analysis of vector sequences x(m) (m\(\geq 0)\) for which \(x(m)\sim \sum \nu (j)\lambda (j)^ m\) (0\(\leq j\leq \infty)\) for large m, where the \(\nu\) (j) are linearly independent vectors in a normed linear space \({\mathfrak B}\) and the \(\lambda\) (j) are scalars for which \(| \lambda (1)| \geq | \lambda (2)| \geq....\) Use is made of determinantal expressions G, F and T defined in terms of a double sequence \(\mu\) and a simple sequence b. With m,n\(\geq 0\), \(k\geq 1\), G(\(\mu|\) m,n;k) is the determinant formed from \(\mu (m+i-1,n+j-1)\) (1\(\leq i,j\leq k)\), F(b,\(\mu| m,n;k)\) is formed from the row \(b(n+j-1)\) \((1\leq j\leq k+1)\) followed by the first k rows of \(G(\mu | m,n;k+1)\), and T(b,\(\mu| m,n;k)\) is the quotient \(F(b,\mu | m,n;k)/G(\mu | m,n+1;k)\). Q(1),Q(2),... being linearly independent bounded linear functionals on \({\mathfrak B}\), \(\mu\) can be defined by setting \(\mu (m,n)=Q(m+1| \quad x(n)).\)
It is found that quotients T of contiguous orders satisfy three term recurrence relationships containing a coefficient d(n,k) which, subject to suitable conditions, tends to \(\lambda\) (k) as n increases. Again, a single functional Q may be used to define \(\mu (m,n)=Q(x(m+n))\) and a similar result holds. Further processes of the same type are examined.