Type II Hermite-Padé approximations of generalized hypergeometric series
The author studies simultaneous Hermite-Padé approximation to generalized hypergeometric and \(q\)-hyper\-geo\-metric series of the form \[ F(t)=\sum_{n=0}^{\infty}{\prod_{k=0}^{n-1}P(k)\over\prod_{k=0}^{n-1}Q(k)}t^n,\;F_q(t)=\sum_{n=0}^{\infty}{\prod_{k=0}^{n-1}P(q^k)\over\prod_{k=0}^{n-1}Q(q^k)}t^n, \] where \(P\) and \(Q\) are polynomials. The main method used to prove the results is based on a paper by \textit{W. Maier} [J. f. M. 156, 93--148 (1927; JFM 53.0340.02)] and a recent modification introduced by the author [J. Math. Soc. Japan 61, No. 1, 291--213 (2009; Zbl 1169.11031)]. The interest in the type of results given, originates from the study of simultaneous approximation in the context of algebraic independence and irrationality in the Theory of Numbers. After a short introduction there are four sections dedicated to {\parindent=6mm \begin{itemize}\item[A.] Hypergeometric functions \(F\), with its main result the explicit form of the type II Hermite-Padé approximants in the variable \(t\) for the \(d\) series \(\theta^bF(t),0\leq b\leq d-1\) at \(m\) points (for \(d=1\) the theorem gives the type II simultaneous approximants). NB. \(\theta=t {d\over dt}\). \end{itemize}} Furthermore, several applications are given (classical hypergeometric seris, exponential series - these are actually \({}_1F_1(1;a;z)\) series - , logarithmic and polylogarithmic series) {\parindent=6mm \begin{itemize}\item[B.] The remainder series technique (using the modified Maier technique), leading to diagonal Hermite-Padé approximants with a free parameter. \item[C.] \(q\)-hypergeometric functions \(F_q\) as given above, giving a new proof of a theorem due to \textit{Th. Stihl} [Math. Ann. 268, 21--41 (1984; Zbl 0519.10024)]. \item[D.] A new proof (using the modified Stihl-Maier method) for a result due to the author in a previous paper [Zbl 1169.11031]. \end{itemize}} The paper concludes with an appendix with some results on Stirling numbers (needed for the proofs) and a list of 18 references.
scientific article; zbMATH DE number 6004436
scientific article