so that the two-term

recurrence relation (51) can be rewritten as ([c.sub.0] = 1)

Proof: Upon extending [u.sub.n] to the negative integers using the

recurrence relation [u.sub.-4] = [3u.sub.n-2] - [u.sub.n], we note that [u.sub.n] = -[u.sub.-n-2].

For each fixed m, these two generating functions share the same denominator, hence the same

recurrence relation. We used Pascal programs and Mathematica 6 to carry out the computation.

A

recurrence relation for marginal moment generating function for lgos from df (1.5) can be obtained in the following theorem.

Then method of lines, semidiscritization approach, is used to transform the model partial differential equation into a system of first-order linear ordinary differential equations whose solution satisfies a

recurrence relation involving matrix exponential function.

Moreover, combining the annihilation equation and the recursion formula (6) we obtain the following

recurrence relation.

Firstly find the

recurrence relation for coefficients and construct with it a program using the calculator to automatically generate coefficients of the series.

** The sequence [{[B.sub.n]}.sub.n[greater than or equal to]0] satisfies the three-term

recurrence relationIt is known that the

recurrence relation for incomplete gamma functions {[Lambda] (a + n, x)}, 0 [is less than or equal to] a [is less than] 1, n = 0, 1,2, ..., when x is positive, is unstable--more so the larger x.

Then the

recurrence relation (2.25) can be expressed as

After studying it carefully, it is found that the solutions cannot be given exactly due to the complicated three-term

recurrence relation. The method used by him is nothing but the Bethe ansatz method as summarized in [13].

Then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the characteristic polynomial which determines the

recurrence relation that [h.sub.m](n) satisfies.