Transcendence of some multivariate power series

Abstract

We prove some transcendence results for the sums of some multivariate series of the form {\sum }_{j\mathrm{1},j\mathrm{2},\cdots jm=\mathrm{0}}^{\infty }{C}_{j\mathrm{1}j\mathrm{2}\cdots jm}{(r_{\mathrm{1}}^{j\mathrm{1}}{r}_{\mathrm{2}}^{j\mathrm{2}}\cdots r_m^{jm})}^n for n = 1,2, where C_{j\mathrm{1}j\mathrm{2}\cdots jm} are some rational functions of j\mathrm{1}+j\mathrm{2}+\cdots jm.

Keywords

Transcendental number / multivariate power series

References

[1] BakerA. Transcendental Number Theory. Cambridge: Cambridge University Press, 1975

[2] BundschuhP, ZhouP. Arithmetical results on certain multivariate power series. Bull Lond Math Soc, 2006, 38(2): 192-200

[3] ZhouP, CuytA, TanJ. General order multivariate Padé approximants for pseudomultivariate functions II. Math Comp, 2009, 78(268): 2137-2155

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2014 Higher Education Press and Springer-Verlag Berlin Heidelberg