Product-Difference Recursion Algorithms

In Section V, we have formally provided simple expressions [Eqs. (5.14), (5.15), and (5.16)] that allow passing from the moments to the parameters of the continued fractions. From a purely algebraic point of view the situation is satisfactory, but not from an operative point of view, an aspect which has often been overlooked in the literatiu e. Indeed, formulas bt ed on Hankel determinants could hardly be used for steps up to it == 10, because of numerical instabilities inherent in the moment problem. On the other hand, in a variety of physical problems (typical are those encountered in solid state physics ), the number of moments practically accessible may be several tens up to 100 or so the same happens in a number of simulated models of remarkable interest in determining the asymptotic behavior of continued fractions. In these cases, more convenient algorithms for the economical evaluation of Hankel determinants must be considered. But the point to be stressed is that in any case one must know the moments with a 

A Hankel determinant D is a function of 2n +1 independent parameters (the moments) yet when constructed explicitly it requires a matrix with (ra -fl) elements. The problem of finding efficient algorithms, which take into account the peculiar persymmetric structure of the Hankel matrices [left diagonals of (S.13) are formed with the same element], has been considered in the literature by several authors. We discuss here in detail a recent satisfactory solution of this problem, obtained within the memory function formalism, and then compare it with other algorithms. 

Using the standard memory function techniques (Chapters I and IV), we have that Oo(l) satisfies the Volterra integro-differential equation 

It is convenient to expand the memory function Oi(t) into powers of t  

Substituting the power expansion (6.1), (6.3), and (6.4) into (6.2) and comparing term by term the coefficients with the same power of t, we obtain the following simple expressions of Oq, b and 

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