The sign regularity of the auxiliary family \(g_ i(x) = x^{\alpha_ i}(-\ln x)^{\beta_ i}\) in convergence acceleration processes using the E-algorithm (Q1919426)

In the following \(N\geq 0\) is an integer, \(g(x|\tau)\) \((0\leq \tau\leq N)\) are suitable real valued functions of the real variable \(x, {\mathcal D}\) is \(d/dx\) and, with \(0\leq n\leq N\), the Wronskian \(W(x,g |n)\) is the determinant of the matrix whose \((\tau+1)^{\text{th}}\) row and \((\upsilon+1)^{\text{th}}\) column element is \({\mathcal D}^\upsilon g(x |\tau)\) \((0\leq \tau, \upsilon\leq n)\). It is shown that \(W\) may be constructed by use of a recursive process involving functions \(f(x,k|\tau)\) \((0\leq k\leq N,\;0\leq \tau\leq N-k)\) in which initially \(f(x,0 |\tau)=g(x |\tau)\) \((0\leq\tau\leq N)\) and thereafter \[ f\bigl(x,k+1 |\tau\bigr) = {\mathcal D} \biggl\{ f\bigl (x,k |\tau+1\bigr)/f \bigl(x,k |0 \bigr) \biggr\} \] \((0\leq k<N,\;0\leq \tau<N-k)\); then \[ W \bigl(x,g |n\bigr) = \biggl\{\prod f \bigl(x,k |0\bigr)^{n-k+1} |0\leq k\leq n \biggr\} \] for \(0\leq n \leq N\). The author gives general expressions for the functions \(f\) (and hence for \(W)\) in the special cases in which the \(g\) have the forms (*) \(g(x |\tau) = x^\alpha \{-\ln (x)\}^{\beta (\tau)}\) and \(g(x|\tau) = x^{\alpha (\tau)}\) (he gives a formula (p. 241, l. 18) relating to the case in which \(g(x|\tau) = \{-\ln(x)\}^{\beta (\tau)}\) which is not obtained from (*) by setting \(\alpha=0)\). He works out, with the aid of a symbol manipulating computer program, the first few \(f\) relating to the case in which \(g(x|\tau) = x^{\alpha (\tau)}\{-\ln (x)\}^{\beta (\tau)}\) (expressions of this form feature in the error analysis of certain numerical methods for the solution of differential equations). Known theory concerning Wronskians \(W(x,g |n)\) and determinants of matrices whose elements are \(g[x(\upsilon) |\tau]\), where \(x(\upsilon)\) \((0\leq \upsilon\leq N)\) is a prescribed sequence, is used to determine the sign of such determinants (the latter feature in academic studies of certain extrapolation processes).