A look-ahead strategy for the implementation of some old and new extrapolation methods (Q1911439)

This paper may be perceived as an essay on the contraction of continued fractions. \(a(\omega)\), \(b(\omega)\) \((\omega\geq 1)\) being a prescribed sequence of pairs of members of a field \(\mathfrak K\), it is possible to compute the members \(p(\omega)\) \((\omega\geq 1)\) of a left module \(\mathfrak M\) over \(\mathfrak K\) from the initial values \(p(- 1)\), \(p(0)\) by use of a recursion of the form \[ p(\omega+ 1)= b(\omega+ 1) p(\omega)+ a(\omega+ 1) p(\omega- 1)\tag{\(*\)} \] for \(\omega\geq 0\). Subject to certain conditions, \(a(\omega)\), \(b(\omega)\) \((\omega\geq 1)\) may be determined from a sequence \(p(\omega)\) \((\omega\geq -1)\) in \(\mathfrak M\) by use of this recursion. \(\omega(k)\) \((k\geq 0)\) being a strictly increasing sequence of positive integers it is, subject to further conditions, possible by elimination to determine the coefficients in a three term recursion involving \(p\{\omega(k+ 1)\}\), \(p\{\omega(k)\}\) and \(p\{\omega(k- 1)\}\), as in the contraction of continued fractions. Mathematical objects belonging to various systems may, subject to suitable existence conditions, be represented as determinantal quotients; they satisfy recursions of the form \((*)\) in which \(a\), \(b\) are also expressible as determinantal quotients. This is true, in particular, of polynomials satisfying orthogonality conditions; in this case \(\mathfrak M\) is the algebraic extension of \(\mathfrak K\). The theory can be extended to the case in which the existence conditions break down and only the quotients defining \(p\{\omega(k)\}\) \((k\geq 0)\) are meaningful, and also to the case in which, taking \(\mathfrak K\) to be \(\mathbb{C}\) for example, only these quotients are numerically well determined; in such a case the quotient expressions for \(a\) and \(b\) are similarly vitiated. A three term recursion involving \(p\{\omega(k)\}\) \((k\geq 0)\) may still be derived as, for example, in the construction of an associated continued fraction when a corresponding continued fraction is undefined.