Some extensions of M-fractions related to strong Stieltjes distributions (Q1591298)

Two-points Padé approximants correspondinq to series expansions at the origin and the point at infinity can be derived as convergents of suitable continued fractions now generally known as \(M\)-fractions [see \textit{J. H. Cabe}, J. Inst. Math. Appl. 15, 363-372 (1975; Zbl 0303.40003)]. The present paper studies even and odd extensions of these \(M\)-fractions [by an even (odd) extension of a continued fraction it should be understood a continued fraction whose even (odd) order convergents are the successive convergents of the original continued fraction] connected with the series expansions determined by strong Stieltjes distributions; in particular the so-called \(S^3[\omega,\beta,b]\) whose elements satisfy the symmetric property \[ d\psi(t)/t^\omega =-d\psi (\beta^2t^{-1})/ (\beta^2 t^{-1})^\omega \quad \text{where} \quad 0<\beta <b\leq\infty, \] \(a=b^{-1} \beta^2\), \(2\omega \in\mathbb{Z}\) for \(t\in(a,b)\) [see \textit{C. F. Bracciali}, \textit{J. H. McCabe} and \textit{A. Sri Ranga}, J. Comput. Appl. Math. 105, No. 1-2, 187-198 (1999)] are specially discussed. The authors extend results given in earlier papers by \textit{J. H. McCabe} [Padé approximation and its applications, Proc. Conf. Amsterdam 1980, Lect. Notes Math. 888, 290-299 (1981; Zbl 0469.41014)] and \textit{W. B. Jones}, \textit{O. Njåstad} and \textit{W. J. Thron} [Constructive Approximation 2, 197-211 (1986; Zbl 0634.41015)]. Some properties regarding the coefficients in the even and odd extensions when the distribution is assumed to be an \(S^3[\omega, \beta,b]\)-distribution, are obtained. Some specific examples are also discussed.