We first describe properties and recursion formulas for this method and subsequently explore simplifications that arise when A is symmetric.

Following Heyouni [12], the recursion formulas for the extended global Arnoldi method can be written as

The result follows from the recursion formulas of Algorithm 1.

The following result shows the relation between leading and trailing coefficients and the coefficients in the recursion formulas of Proposition 2.2.

Let the sequence of orthonormal Laurent polynomials be defined by the recursion formulas of Proposition 2.2, and let the [[alpha].sub.j], [[beta].sub.j], and [[delta].sub.j] be the recursion coefficients.

Substituting (38) into (37), we raise the

recursion formulas for [k.sub.j] and [g.sub.j]:

In Section 2.1, we introduce the recursion formulas of coefficients of orders 1 and 2, which are usually used, and some properties are given.

In this subsection, we present the recursion formulas of [[??].sub.p,l] (p = 7, 8, 9, 10) for reference.

In this paper, we propose recursion formulas to compute the coefficients of the fractional linear multistep schemes.

Proof: This can be proved using

recursion formulas derived from the urn model.

where [B.sub.q](k) is given by the recursion formulas: [B.sub.q](0) = 1/q-1,

[B.sub.q](0) = [[infinity].summation over (m=1)] [1/[q.sup.m]] = 1/q - 1, and [B.sub.q](k) satisfies the recursion formula