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Extension of Some Theorems of W. Schwarz

  • Michael Coons (a1)

Abstract

In this paper, we prove that a non–zero power series $F(z\text{)}\in \mathbb{C}\text{ }[[z]]$ satisfying

$$F({{z}^{d}})\,=\,F(z)\,+\,\frac{A(z)}{B(z)},$$

where $d\,\ge \,2,\,A(z),\,B(z)\,\in \,\mathbb{C}[z]$ , with $A(z)\,\ne \,0$ and $\deg \,A(z),\,\deg \,B(z)\,<\,d$ is transcendental over $\mathbb{C}(z)$ . Using this result and a theorem of Mahler’s, we extend results of Golomb and Schwarz on transcendental values of certain power series. In particular, we prove that for all $k\,\ge \,2$ the series ${{G}_{k}}(z):=\mathop{\sum }_{n=0}^{\infty }{{z}^{{{k}^{n}}}}{{(1-{{z}^{{{k}^{n}}}})}^{-1}}$ is transcendental for all algebraic numbers $z$ with $\left| z \right|\,<\,1$ . We give a similar result for ${{F}_{k}}(z):=\mathop{\sum }_{n=0}^{\infty }{{z}^{{{k}^{n}}}}{{(1+{{z}^{{{k}^{n}}}})}^{-1}}$ . These results were known to Mahler, though our proofs of the function transcendence are new and elementary; no linear algebra or differential calculus is used.

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References

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[1] Duverney, D., Transcendence of a fast converging series of rational numbers. Math. Proc. Cambridge Philos. Soc. 130(2001), no. 2, 193207. doi:10.1017/S0305004100004783
[2] Duverney, D. and Nishioka, K., An inductive method for proving the transcendence of certain series. Acta Arith. 110(2003), no. 4, 305330. doi:10.4064/aa110-4-1
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[7] Mahler, K., Über das Verschwinden von Potenzreihen mehrerer Ver änderlicher in speziellen Punktfolgen. Math. Ann. 103(1930), no. 1, 573587. doi:10.1007/BF01455711
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[10] Nishioka, Kumiko, Mahler Functions and Transcendence. Lecture Notes in Mathematics, 1631, Springer-Verlag, Berlin, 1996.
[11] Schwarz, W., Remarks on the irrationality and transcendence of certain series. Math. Scand 20(1967), 269274.
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