Rational approximations to\( _ 1F_ 1(l,c,az)\) in (0,\(\infty)\) (Q579557)

In Trans. Am. Mat. Soc. 235, 395-402 (1978; Zbl 0386.41010), \textit{Q. I. Rahman} and \textit{G. Schmeisser} conjectured that \(\lim (\rho_ n)^{1/n}=1/9\) where \(\rho_ n=\min_{\Pi_ n}\| e^{-z}-\pi (z)\|_{C[0,\infty)}\), where \(\pi (z)\in \Pi_ n\) the set of polynomials of degree \(<n\). It was later conjectured by \textit{A. Schönhage}, SIAM J. Numer. Anal. 19, 1067-1080 (1982; Zbl 0514.41018) that 1/9 may be replaced by a smaller constant. In this paper the author proves that \[ \lim (\min_{\Pi_ n}\|_ 1F_ 1(1,c,-z)-\pi_ n(z))\|_{C[0,\infty)})^{1/n}<1/9, \] where \(c\in {\mathbb{R}}\setminus \{0,-1,-2,...\}\). Since \({}_ 1F_ 1(1,1,-z)=e^{-z}\), this includes the conjecture of Schönhage. The proof is based on a series of lemmas involving the gamma and beta functions.